Properties

 Label 33.3.g.a Level 33 Weight 3 Character orbit 33.g Analytic conductor 0.899 Analytic rank 0 Dimension 16 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 33.g (of order $$10$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.899184872389$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + 1868 x^{7} - 1619 x^{6} - 16804 x^{5} + 32427 x^{4} + 43316 x^{3} - 71672 x^{2} + 83521$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{2} -\beta_{14} q^{3} + ( -\beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{15} ) q^{4} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} - 3 \beta_{15} ) q^{5} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{6} + ( -3 + 2 \beta_{1} + 2 \beta_{4} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} ) q^{7} + ( -\beta_{2} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} + 3 \beta_{15} ) q^{8} + 3 \beta_{8} q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{2} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{2} -\beta_{14} q^{3} + ( -\beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{15} ) q^{4} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} - 3 \beta_{15} ) q^{5} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{6} + ( -3 + 2 \beta_{1} + 2 \beta_{4} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} ) q^{7} + ( -\beta_{2} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} + 3 \beta_{15} ) q^{8} + 3 \beta_{8} q^{9} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} + 6 \beta_{14} + 6 \beta_{15} ) q^{10} + ( -6 - \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + \beta_{6} - 10 \beta_{7} - 9 \beta_{8} + \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{11} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{5} + 3 \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{12} + ( 3 + \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{8} - \beta_{10} - 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{13} + ( -4 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{5} + 3 \beta_{6} - \beta_{7} - 2 \beta_{8} - 6 \beta_{10} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 4 \beta_{14} + 8 \beta_{15} ) q^{14} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} + 5 \beta_{8} + \beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{15} + ( 3 - 6 \beta_{1} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} - 4 \beta_{14} - 7 \beta_{15} ) q^{16} + ( -1 + 3 \beta_{3} + 5 \beta_{4} + \beta_{7} + 6 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - \beta_{13} + 3 \beta_{15} ) q^{17} + ( -3 + 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} ) q^{18} + ( -1 + 3 \beta_{1} + 5 \beta_{2} - 8 \beta_{3} + 6 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 5 \beta_{10} - 2 \beta_{12} - 5 \beta_{14} - 5 \beta_{15} ) q^{19} + ( 4 - 3 \beta_{1} + 3 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 8 \beta_{6} - \beta_{7} - 9 \beta_{8} - 4 \beta_{9} + 8 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 12 \beta_{14} - 14 \beta_{15} ) q^{20} + ( 3 + 6 \beta_{2} + \beta_{3} + 6 \beta_{4} - \beta_{6} + 5 \beta_{7} + 5 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{14} + \beta_{15} ) q^{21} + ( 10 + 2 \beta_{1} + 3 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} + 7 \beta_{6} + 16 \beta_{7} + 20 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} - 4 \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{22} + ( 8 - \beta_{1} + \beta_{2} - \beta_{4} + 3 \beta_{5} - 3 \beta_{7} - \beta_{9} + 5 \beta_{10} - \beta_{11} + 2 \beta_{13} - 6 \beta_{14} - \beta_{15} ) q^{23} + ( 2 - \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 7 \beta_{7} - 4 \beta_{8} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{24} + ( -2 \beta_{1} + 2 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} - 5 \beta_{8} + 2 \beta_{9} - 11 \beta_{10} + 2 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} + 15 \beta_{15} ) q^{25} + ( 3 + \beta_{1} + 4 \beta_{2} - 11 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - 10 \beta_{15} ) q^{26} -3 \beta_{15} q^{27} + ( -1 + \beta_{2} + 11 \beta_{3} - 7 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - 7 \beta_{8} - 6 \beta_{10} + 2 \beta_{13} - 7 \beta_{14} + 2 \beta_{15} ) q^{28} + ( 17 - 3 \beta_{1} + \beta_{2} + 7 \beta_{3} - 2 \beta_{4} - 12 \beta_{5} - \beta_{6} + 7 \beta_{7} + 2 \beta_{8} - \beta_{9} - 8 \beta_{10} - \beta_{12} + \beta_{13} + 15 \beta_{14} + 7 \beta_{15} ) q^{29} + ( -4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} - 16 \beta_{7} - 22 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{12} - \beta_{14} + \beta_{15} ) q^{30} + ( -2 + 3 \beta_{1} - 6 \beta_{2} - 7 \beta_{3} - 12 \beta_{4} - 7 \beta_{5} + 18 \beta_{6} + \beta_{7} + 4 \beta_{8} + 7 \beta_{10} - 2 \beta_{11} - 2 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} ) q^{31} + ( 8 + \beta_{2} + 6 \beta_{3} + \beta_{4} - 17 \beta_{5} + \beta_{6} + 17 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + 14 \beta_{14} + 13 \beta_{15} ) q^{32} + ( -1 + 2 \beta_{1} - \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 12 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} + 4 \beta_{14} + 5 \beta_{15} ) q^{33} + ( -21 + 4 \beta_{1} + 3 \beta_{2} - \beta_{3} + 5 \beta_{4} + \beta_{5} - 4 \beta_{6} + 3 \beta_{7} + 4 \beta_{8} + 13 \beta_{10} - 13 \beta_{14} ) q^{34} + ( -23 + 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 17 \beta_{5} + 9 \beta_{7} - 7 \beta_{8} + 4 \beta_{10} + 2 \beta_{12} + 8 \beta_{14} - 6 \beta_{15} ) q^{35} + ( 3 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{6} + 6 \beta_{8} - 3 \beta_{10} + 3 \beta_{14} + 3 \beta_{15} ) q^{36} + ( -7 - 11 \beta_{1} - 12 \beta_{2} + \beta_{3} - \beta_{4} - 9 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} - 5 \beta_{8} - 2 \beta_{9} - 7 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{37} + ( -23 + 16 \beta_{1} - 2 \beta_{2} + \beta_{3} + 5 \beta_{4} + 20 \beta_{5} + 4 \beta_{6} - 11 \beta_{7} - 6 \beta_{8} + 3 \beta_{9} + 6 \beta_{12} - 3 \beta_{13} + 4 \beta_{14} + 11 \beta_{15} ) q^{38} + ( -1 - 4 \beta_{2} + \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} + 9 \beta_{7} - 3 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{13} - 2 \beta_{14} ) q^{39} + ( 1 + 7 \beta_{1} - 2 \beta_{2} + 9 \beta_{3} + 5 \beta_{4} + 14 \beta_{5} + 12 \beta_{6} + 9 \beta_{7} + 32 \beta_{8} + 2 \beta_{9} - 7 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + 12 \beta_{14} + 5 \beta_{15} ) q^{40} + ( -1 - 7 \beta_{1} - 5 \beta_{2} - 17 \beta_{3} - 7 \beta_{4} - 10 \beta_{5} + 8 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} + 12 \beta_{10} + 5 \beta_{11} - 2 \beta_{12} - 12 \beta_{14} - 12 \beta_{15} ) q^{41} + ( -16 - \beta_{1} + 5 \beta_{2} - 5 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} - 10 \beta_{6} - 18 \beta_{7} - 6 \beta_{8} - 2 \beta_{9} + 5 \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - 5 \beta_{14} - 6 \beta_{15} ) q^{42} + ( -17 - 2 \beta_{1} - 11 \beta_{2} + 5 \beta_{3} - 11 \beta_{4} + 17 \beta_{5} - 6 \beta_{6} - 23 \beta_{7} - 40 \beta_{8} - 4 \beta_{9} - \beta_{10} + 4 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + 5 \beta_{14} + 6 \beta_{15} ) q^{43} + ( -9 - 8 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 20 \beta_{5} - 11 \beta_{6} - 8 \beta_{7} - 17 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - \beta_{12} + 5 \beta_{13} - 7 \beta_{14} - 12 \beta_{15} ) q^{44} + ( -3 - 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} + 3 \beta_{14} + 3 \beta_{15} ) q^{45} + ( 6 + \beta_{1} + 5 \beta_{2} - 6 \beta_{3} - 8 \beta_{5} + \beta_{7} + 5 \beta_{8} - 6 \beta_{10} + 6 \beta_{11} - 2 \beta_{12} - 6 \beta_{13} + 5 \beta_{14} - 7 \beta_{15} ) q^{46} + ( -8 - \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 7 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} + 10 \beta_{7} - \beta_{9} - 9 \beta_{10} - 3 \beta_{11} - 7 \beta_{12} + 8 \beta_{13} - 9 \beta_{14} + \beta_{15} ) q^{47} + ( 3 + 9 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} - 6 \beta_{5} + 3 \beta_{6} + 15 \beta_{7} + 12 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + 3 \beta_{12} - 3 \beta_{13} + 3 \beta_{14} + 2 \beta_{15} ) q^{48} + ( 17 - 8 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 8 \beta_{4} - 15 \beta_{5} + 5 \beta_{6} - 2 \beta_{8} + 4 \beta_{9} - 6 \beta_{10} - 12 \beta_{11} - 4 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{49} + ( 17 + \beta_{2} + 15 \beta_{3} - 2 \beta_{4} + 17 \beta_{5} - \beta_{6} - 25 \beta_{7} + 10 \beta_{8} - 13 \beta_{9} - 11 \beta_{10} - 4 \beta_{14} - \beta_{15} ) q^{50} + ( -4 - 12 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 10 \beta_{4} + 9 \beta_{5} - 4 \beta_{6} - 9 \beta_{7} - 9 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - \beta_{14} - 5 \beta_{15} ) q^{51} + ( 8 + 13 \beta_{1} + 9 \beta_{2} + 4 \beta_{3} + 11 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 9 \beta_{7} + 10 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} + 5 \beta_{14} + 12 \beta_{15} ) q^{52} + ( 25 + 3 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 8 \beta_{4} + 9 \beta_{5} - 2 \beta_{6} + 36 \beta_{7} + 10 \beta_{8} + 16 \beta_{9} + 2 \beta_{10} - 8 \beta_{11} + 8 \beta_{12} - 8 \beta_{13} + 10 \beta_{14} + 18 \beta_{15} ) q^{53} + ( 3 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - 3 \beta_{11} + 3 \beta_{14} + 3 \beta_{15} ) q^{54} + ( 15 + 3 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} + 5 \beta_{5} + \beta_{6} + 7 \beta_{7} - 8 \beta_{8} - 2 \beta_{9} + 23 \beta_{10} + 10 \beta_{11} + 12 \beta_{12} + 2 \beta_{13} - 15 \beta_{14} - 12 \beta_{15} ) q^{55} + ( 13 - 8 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} - 10 \beta_{4} + 20 \beta_{5} + 10 \beta_{6} - 30 \beta_{7} - 10 \beta_{8} + 2 \beta_{9} + 5 \beta_{10} + 2 \beta_{11} - 3 \beta_{12} - 7 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{56} + ( 18 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 24 \beta_{5} + 12 \beta_{7} + 18 \beta_{8} + 2 \beta_{10} - 3 \beta_{11} - 8 \beta_{12} + 3 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} ) q^{57} + ( 39 - 17 \beta_{1} + 25 \beta_{2} - 8 \beta_{3} + 13 \beta_{4} - 16 \beta_{5} - 20 \beta_{6} + 13 \beta_{7} + 27 \beta_{8} - 8 \beta_{9} + 6 \beta_{10} - 8 \beta_{12} + 16 \beta_{13} - 9 \beta_{14} - 22 \beta_{15} ) q^{58} + ( 13 \beta_{1} + 7 \beta_{2} + 17 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} - 20 \beta_{7} - 3 \beta_{8} - 7 \beta_{9} - 8 \beta_{10} + 6 \beta_{11} + \beta_{12} - 7 \beta_{13} + \beta_{14} + 11 \beta_{15} ) q^{59} + ( 25 + 10 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 24 \beta_{5} + 9 \beta_{6} + 38 \beta_{7} + 44 \beta_{8} + \beta_{9} - 2 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} + 7 \beta_{15} ) q^{60} + ( 2 + 17 \beta_{2} - 7 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} - 17 \beta_{6} + 14 \beta_{7} - 12 \beta_{8} - 2 \beta_{10} - 4 \beta_{13} + 13 \beta_{14} + 3 \beta_{15} ) q^{61} + ( -6 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} - 10 \beta_{4} - 39 \beta_{5} - 5 \beta_{7} - 49 \beta_{8} + 4 \beta_{9} + 8 \beta_{10} + \beta_{11} + 5 \beta_{12} - 4 \beta_{13} - 11 \beta_{14} - 12 \beta_{15} ) q^{62} + ( 3 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} - 9 \beta_{7} - 9 \beta_{8} - 6 \beta_{9} - 6 \beta_{12} - 3 \beta_{14} - 6 \beta_{15} ) q^{63} + ( -12 + 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} + 12 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} + 8 \beta_{10} + 6 \beta_{11} + 2 \beta_{12} + 6 \beta_{13} - 6 \beta_{14} - 4 \beta_{15} ) q^{64} + ( 32 + 2 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} + 9 \beta_{4} - 27 \beta_{5} - 4 \beta_{6} + 33 \beta_{7} + 60 \beta_{8} - 6 \beta_{9} - 7 \beta_{10} + 6 \beta_{11} + \beta_{12} - \beta_{13} - 9 \beta_{14} - 2 \beta_{15} ) q^{65} + ( -4 - 5 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} - 9 \beta_{4} + 21 \beta_{5} - 7 \beta_{6} - 8 \beta_{7} - 17 \beta_{8} - 4 \beta_{9} - \beta_{10} + 5 \beta_{11} - 5 \beta_{12} + 2 \beta_{13} - 11 \beta_{14} - 15 \beta_{15} ) q^{66} + ( -5 + 13 \beta_{1} - 7 \beta_{2} + 25 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} + 9 \beta_{8} + 4 \beta_{9} - 11 \beta_{10} + 4 \beta_{11} + 6 \beta_{12} - 2 \beta_{13} + 15 \beta_{14} + 4 \beta_{15} ) q^{67} + ( -21 - 9 \beta_{1} - 20 \beta_{2} - 15 \beta_{3} + 29 \beta_{5} - 29 \beta_{7} - 26 \beta_{8} - 15 \beta_{10} + \beta_{11} - 7 \beta_{12} - \beta_{13} - 9 \beta_{14} + 2 \beta_{15} ) q^{68} + ( 15 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} + 5 \beta_{6} + 2 \beta_{7} + 20 \beta_{8} - \beta_{9} + 4 \beta_{10} + \beta_{11} + \beta_{12} - 10 \beta_{14} - 4 \beta_{15} ) q^{69} + ( -10 - 13 \beta_{1} - 26 \beta_{2} - 17 \beta_{3} - 13 \beta_{4} + 28 \beta_{5} + 3 \beta_{6} - 34 \beta_{7} - 48 \beta_{8} + 6 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} - 4 \beta_{12} + 6 \beta_{13} - 4 \beta_{14} - 15 \beta_{15} ) q^{70} + ( -43 - 10 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - 8 \beta_{4} + 41 \beta_{5} + 10 \beta_{6} - 33 \beta_{7} - 40 \beta_{8} + 3 \beta_{9} - \beta_{10} - 2 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} - 13 \beta_{15} ) q^{71} + ( -3 + 6 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} + 9 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 3 \beta_{13} - 6 \beta_{14} - 3 \beta_{15} ) q^{72} + ( -29 + 4 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} + 10 \beta_{4} + 22 \beta_{5} + 7 \beta_{6} - 6 \beta_{7} + 10 \beta_{8} - 6 \beta_{9} - 13 \beta_{10} - 8 \beta_{11} - 14 \beta_{12} + 6 \beta_{13} + 12 \beta_{14} + 2 \beta_{15} ) q^{73} + ( 1 + 2 \beta_{1} + 5 \beta_{2} + 25 \beta_{3} + 6 \beta_{4} - 29 \beta_{5} - 4 \beta_{6} + 72 \beta_{7} + 76 \beta_{8} - 3 \beta_{9} - 11 \beta_{10} - 8 \beta_{11} - 3 \beta_{12} + 15 \beta_{14} + 19 \beta_{15} ) q^{74} + ( -40 - 3 \beta_{1} - 7 \beta_{2} + 10 \beta_{3} - 8 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} - 40 \beta_{7} - 7 \beta_{8} + 6 \beta_{9} - 10 \beta_{10} - 7 \beta_{11} + 3 \beta_{12} - 7 \beta_{13} + 13 \beta_{14} + 16 \beta_{15} ) q^{75} + ( -29 + 6 \beta_{1} - 11 \beta_{2} - 5 \beta_{3} - 11 \beta_{4} + 37 \beta_{5} - 2 \beta_{6} - 23 \beta_{7} - 60 \beta_{8} - 8 \beta_{9} + 11 \beta_{10} + 8 \beta_{11} + 10 \beta_{12} - 10 \beta_{13} - 7 \beta_{14} - 18 \beta_{15} ) q^{76} + ( -11 + 22 \beta_{1} + 14 \beta_{2} - 10 \beta_{3} + 23 \beta_{4} - 21 \beta_{5} - 6 \beta_{6} - 8 \beta_{7} + 41 \beta_{8} - 9 \beta_{9} + 10 \beta_{10} - 3 \beta_{11} - 9 \beta_{12} + 9 \beta_{13} - 7 \beta_{14} + 5 \beta_{15} ) q^{77} + ( -8 - 6 \beta_{1} - 5 \beta_{2} - \beta_{3} - 9 \beta_{4} - 33 \beta_{5} + 7 \beta_{6} + 26 \beta_{7} - 7 \beta_{8} + \beta_{9} + 5 \beta_{10} + \beta_{11} - 2 \beta_{13} - 4 \beta_{14} + \beta_{15} ) q^{78} + ( 31 - 5 \beta_{1} + \beta_{2} + 18 \beta_{3} - 38 \beta_{5} + 10 \beta_{7} + 16 \beta_{8} + 18 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} - \beta_{14} + 5 \beta_{15} ) q^{79} + ( -74 - 7 \beta_{1} + 2 \beta_{2} + \beta_{3} - 16 \beta_{4} + 30 \beta_{5} - 13 \beta_{6} - 12 \beta_{7} - 88 \beta_{8} + 5 \beta_{9} + 21 \beta_{10} - 4 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} + 5 \beta_{14} - 19 \beta_{15} ) q^{80} + 9 \beta_{7} q^{81} + ( 14 - 21 \beta_{2} + 21 \beta_{3} - 10 \beta_{4} - 24 \beta_{5} - \beta_{6} + 14 \beta_{7} + 4 \beta_{8} + 10 \beta_{9} + 20 \beta_{12} - 10 \beta_{13} + 31 \beta_{14} + 59 \beta_{15} ) q^{82} + ( -14 - 19 \beta_{2} - 28 \beta_{3} - 20 \beta_{4} - 24 \beta_{5} + 19 \beta_{6} - 30 \beta_{7} + 32 \beta_{8} + 12 \beta_{9} + 26 \beta_{10} + 10 \beta_{13} - 8 \beta_{14} - 12 \beta_{15} ) q^{83} + ( -26 + 9 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} + \beta_{4} + 33 \beta_{5} + 13 \beta_{6} - 6 \beta_{7} + 21 \beta_{8} + 8 \beta_{9} - 2 \beta_{10} + \beta_{11} + 9 \beta_{12} - 8 \beta_{13} + 13 \beta_{14} + 19 \beta_{15} ) q^{84} + ( -23 - 4 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} - 11 \beta_{4} + 15 \beta_{5} + 7 \beta_{6} - 49 \beta_{7} - 38 \beta_{8} + 8 \beta_{9} - \beta_{10} + 8 \beta_{11} + 8 \beta_{12} - 2 \beta_{14} - 5 \beta_{15} ) q^{85} + ( 77 + 3 \beta_{1} - 4 \beta_{2} + 19 \beta_{3} - 18 \beta_{4} - 19 \beta_{5} + 24 \beta_{6} + 75 \beta_{7} + 46 \beta_{8} - 10 \beta_{9} - 19 \beta_{10} + 7 \beta_{11} - 5 \beta_{12} + 7 \beta_{13} + 23 \beta_{14} + 18 \beta_{15} ) q^{86} + ( -26 + 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 21 \beta_{5} + 5 \beta_{6} - 26 \beta_{7} - 47 \beta_{8} - 12 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 10 \beta_{15} ) q^{87} + ( 8 - 5 \beta_{1} - 4 \beta_{2} - \beta_{3} + 8 \beta_{4} + 17 \beta_{5} + 11 \beta_{6} + 11 \beta_{7} + 31 \beta_{8} + 6 \beta_{9} - 40 \beta_{10} - 12 \beta_{11} - 6 \beta_{13} + 36 \beta_{14} + 32 \beta_{15} ) q^{88} + ( 26 + 13 \beta_{1} + 17 \beta_{2} + 26 \beta_{3} + 3 \beta_{4} - 21 \beta_{5} - 10 \beta_{6} + 31 \beta_{7} + 10 \beta_{8} + 3 \beta_{9} - 19 \beta_{10} + 3 \beta_{11} + 8 \beta_{12} + 2 \beta_{13} + 22 \beta_{14} + 3 \beta_{15} ) q^{89} + ( 9 + 6 \beta_{3} - 12 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} + 6 \beta_{10} + 6 \beta_{12} - 6 \beta_{14} + 12 \beta_{15} ) q^{90} + ( -1 + 14 \beta_{1} - 20 \beta_{2} + 6 \beta_{3} - 17 \beta_{4} + 33 \beta_{5} + 9 \beta_{6} - 17 \beta_{7} + 2 \beta_{8} + 6 \beta_{9} + 17 \beta_{10} + 6 \beta_{12} - 12 \beta_{13} - 5 \beta_{15} ) q^{91} + ( 4 + 5 \beta_{1} + 5 \beta_{2} + 8 \beta_{3} - 10 \beta_{5} + 4 \beta_{6} + 21 \beta_{7} + 18 \beta_{8} + 3 \beta_{9} + 10 \beta_{10} - 11 \beta_{11} + 8 \beta_{12} + 3 \beta_{13} + 8 \beta_{14} + 19 \beta_{15} ) q^{92} + ( 17 + 8 \beta_{1} - 10 \beta_{2} + 7 \beta_{4} - 21 \beta_{5} - 16 \beta_{6} + 5 \beta_{7} + 5 \beta_{8} - 3 \beta_{9} + 7 \beta_{10} + 14 \beta_{11} + 8 \beta_{12} - 4 \beta_{13} - 3 \beta_{14} - 6 \beta_{15} ) q^{93} + ( -26 - 15 \beta_{2} - 5 \beta_{3} - 13 \beta_{4} - 24 \beta_{5} + 15 \beta_{6} + 8 \beta_{7} - 25 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} - 2 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{94} + ( 48 - 3 \beta_{1} + 5 \beta_{2} - 28 \beta_{3} + 2 \beta_{4} - 40 \beta_{5} - 16 \beta_{6} + 17 \beta_{7} - 6 \beta_{8} - 5 \beta_{9} + 2 \beta_{10} - 7 \beta_{11} - 12 \beta_{12} + 5 \beta_{13} - 16 \beta_{14} + 7 \beta_{15} ) q^{95} + ( 3 - 4 \beta_{1} - 3 \beta_{2} + 19 \beta_{3} - 4 \beta_{4} + 18 \beta_{5} - 40 \beta_{7} - 46 \beta_{8} - \beta_{9} - 17 \beta_{10} - \beta_{12} + 9 \beta_{14} + \beta_{15} ) q^{96} + ( -28 + \beta_{1} + 4 \beta_{2} + 23 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} - 33 \beta_{7} - \beta_{8} - 12 \beta_{9} - 23 \beta_{10} - 6 \beta_{12} - 2 \beta_{14} - 8 \beta_{15} ) q^{97} + ( 27 - \beta_{1} + 23 \beta_{2} - 26 \beta_{3} + 23 \beta_{4} - 27 \beta_{5} + 4 \beta_{6} + 31 \beta_{7} + 58 \beta_{8} + 5 \beta_{9} + 22 \beta_{10} - 5 \beta_{11} - 5 \beta_{12} + 5 \beta_{13} - 25 \beta_{14} - 47 \beta_{15} ) q^{98} + ( 6 - 6 \beta_{1} - 15 \beta_{3} + 3 \beta_{4} + 15 \beta_{5} - 3 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + 6 \beta_{10} - 6 \beta_{11} - 3 \beta_{12} + 3 \beta_{13} - 9 \beta_{14} - 9 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 20q^{4} - 4q^{5} - 30q^{7} - 40q^{8} - 12q^{9} + O(q^{10})$$ $$16q + 20q^{4} - 4q^{5} - 30q^{7} - 40q^{8} - 12q^{9} - 10q^{11} - 24q^{12} + 30q^{13} - 2q^{14} - 24q^{15} + 16q^{16} - 10q^{17} - 30q^{18} + 42q^{20} + 42q^{22} + 132q^{23} + 90q^{24} - 2q^{25} + 46q^{26} - 50q^{28} + 160q^{29} + 180q^{30} + 10q^{31} + 12q^{33} - 368q^{34} - 320q^{35} + 60q^{36} - 126q^{37} - 130q^{38} + 30q^{40} - 120q^{41} - 204q^{42} - 206q^{44} - 12q^{45} + 50q^{46} - 150q^{47} - 96q^{48} + 210q^{49} + 330q^{50} - 60q^{51} + 110q^{52} + 342q^{53} + 244q^{55} + 524q^{56} + 60q^{57} + 150q^{58} + 110q^{59} + 36q^{60} - 90q^{61} + 40q^{62} + 90q^{63} - 168q^{64} + 48q^{66} + 36q^{67} + 80q^{68} + 210q^{69} + 340q^{70} - 236q^{71} - 150q^{72} - 350q^{73} - 730q^{74} - 408q^{75} - 390q^{77} - 312q^{78} + 210q^{79} - 806q^{80} - 36q^{81} + 114q^{82} - 190q^{83} - 180q^{84} + 110q^{85} + 736q^{86} + 144q^{88} + 76q^{89} + 60q^{90} + 306q^{91} - 150q^{92} + 144q^{93} - 350q^{94} + 430q^{95} + 450q^{96} - 354q^{97} + 180q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + 1868 x^{7} - 1619 x^{6} - 16804 x^{5} + 32427 x^{4} + 43316 x^{3} - 71672 x^{2} + 83521$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$138203247537734 \nu^{15} - 291050501651580 \nu^{14} - 300665341738367 \nu^{13} - 1705096652769584 \nu^{12} + 9450527129283881 \nu^{11} + 12120978905635508 \nu^{10} - 128500627016425958 \nu^{9} - 144385372422156236 \nu^{8} + 122498246567678379 \nu^{7} + 788622311310230084 \nu^{6} - 133037791520559623 \nu^{5} - 5550683119927089754 \nu^{4} - 1059049981018273091 \nu^{3} + 8378055018470127650 \nu^{2} + 13020347824350349103 \nu - 10624181552957568214$$$$)/ 10999895242048793399$$ $$\beta_{3}$$ $$=$$ $$($$$$-1196620054805 \nu^{15} + 7612440593330 \nu^{14} - 13694873409440 \nu^{13} + 37431650235165 \nu^{12} - 160608059534630 \nu^{11} + 396329540870948 \nu^{10} + 970794484128155 \nu^{9} - 2042076941900150 \nu^{8} + 1143522707363375 \nu^{7} - 3000401551565675 \nu^{6} + 26580224433121053 \nu^{5} + 18238786120480205 \nu^{4} - 74422428601993325 \nu^{3} + 43546075769649555 \nu^{2} + 84006679985420530 \nu + 87548232770773285$$$$)/ 78081122146535134$$ $$\beta_{4}$$ $$=$$ $$($$$$79900318483541295869 \nu^{15} - 414504109012258813809 \nu^{14} + 817712598296562884653 \nu^{13} - 1181236305529453459044 \nu^{12} + 9352590043947181216000 \nu^{11} - 13643260700327505255674 \nu^{10} - 56158309327266771570697 \nu^{9} + 189766153484548192315027 \nu^{8} + 95964394821698805650574 \nu^{7} + 136183765825282943853761 \nu^{6} - 829913842787061972319720 \nu^{5} - 811030725865674829791365 \nu^{4} + 6972042698248130206151734 \nu^{3} + 1663249793505431751688693 \nu^{2} - 10114210902968983105987135 \nu + 5561216538715898968492982$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{5}$$ $$=$$ $$($$$$-122411618273740428915 \nu^{15} + 390636691234340826078 \nu^{14} - 875766486639397095366 \nu^{13} + 1306893009198853700308 \nu^{12} - 11122165667396999800860 \nu^{11} + 1352851863401509576374 \nu^{10} + 68084957445524132210249 \nu^{9} - 180048790704766856126064 \nu^{8} - 7207303915835648139573 \nu^{7} - 246702387055823098949248 \nu^{6} + 488650962940974324993039 \nu^{5} + 1534542665221003166135310 \nu^{4} - 6976139470347708662854381 \nu^{3} + 1584736358268525956493486 \nu^{2} + 6489773955290969592983712 \nu - 5737747263618475322503230$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{6}$$ $$=$$ $$($$$$-136359504183385890890 \nu^{15} + 375227479116253831410 \nu^{14} - 988352396932402617563 \nu^{13} + 1181001191967106814165 \nu^{12} - 10341194991166970946468 \nu^{11} - 5186301744109515584546 \nu^{10} + 72904516698863990088998 \nu^{9} - 223430759423433745295680 \nu^{8} + 152854650725265611821303 \nu^{7} - 104458453399814352710637 \nu^{6} + 131176280566363812419931 \nu^{5} + 1643119706749150800729097 \nu^{4} - 7289423934676201623932819 \nu^{3} + 8681896656673390953560247 \nu^{2} + 7009039219983139174322425 \nu - 14395227503745014111293247$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{7}$$ $$=$$ $$($$$$-170240978662162064119 \nu^{15} + 434330161717510886082 \nu^{14} - 909817832880658011498 \nu^{13} + 1611262423238216663326 \nu^{12} - 14163899845284185510064 \nu^{11} - 6409586754553109280864 \nu^{10} + 91124937595251030901909 \nu^{9} - 159377154247336302510228 \nu^{8} - 25618165363271255976237 \nu^{7} - 382107707960321190084594 \nu^{6} + 269559976623314579423811 \nu^{5} + 2388290245026519865709064 \nu^{4} - 6440011982471751717571965 \nu^{3} - 88831069153588802148432 \nu^{2} + 6767600917775046214158900 \nu - 10633476167661152687325848$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{8}$$ $$=$$ $$($$$$-206074168461557516052 \nu^{15} + 708563165563293316406 \nu^{14} - 1346611723653217936835 \nu^{13} + 2263013632083543127888 \nu^{12} - 18560750721332235068174 \nu^{11} + 5237533698052177949662 \nu^{10} + 133720108959147123348880 \nu^{9} - 287797606597911626958148 \nu^{8} - 55824128700867687245691 \nu^{7} - 399468647443749302413782 \nu^{6} + 979026668318049167132803 \nu^{5} + 3044197524673710725500914 \nu^{4} - 11303824104537481334436817 \nu^{3} + 1845484707680277480548616 \nu^{2} + 11349844369944629860390067 \nu - 12139260995132491579346226$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{9}$$ $$=$$ $$($$$$245204915440680156914 \nu^{15} - 776594441642987820173 \nu^{14} + 2376052169206225011042 \nu^{13} - 4478763958971146181708 \nu^{12} + 24746729416159566188668 \nu^{11} - 11798681872105894888540 \nu^{10} - 77205981106427487557846 \nu^{9} + 358881589762381071470923 \nu^{8} - 401666468771928513101288 \nu^{7} + 725446686209137214367935 \nu^{6} - 711143038597944288083530 \nu^{5} + 239881156011128668612579 \nu^{4} + 11370877427156155063566520 \nu^{3} - 16802658167616863816782085 \nu^{2} + 2853132220575548469672140 \nu + 29383283378807446672223354$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{10}$$ $$=$$ $$($$$$290375359378885003312 \nu^{15} - 743934382221418472242 \nu^{14} + 1767266717836305278230 \nu^{13} - 2850753731085855890079 \nu^{12} + 26153841247373778060042 \nu^{11} + 7902296848980404937094 \nu^{10} - 130800194424547620941876 \nu^{9} + 292119947250793672950606 \nu^{8} + 124867657205700169186062 \nu^{7} + 701839303337308061729221 \nu^{6} - 395644720931754611009464 \nu^{5} - 3190848798199793757451725 \nu^{4} + 11234369776810738073080218 \nu^{3} - 631699645499098517354817 \nu^{2} - 10344016932751524448182256 \nu + 19782979528236067552714073$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{11}$$ $$=$$ $$($$$$-318073994808497231570 \nu^{15} + 2293064170187332819187 \nu^{14} - 4466190043659119653421 \nu^{13} + 7837297929564868783317 \nu^{12} - 37867833616479840448874 \nu^{11} + 105998876476806512378294 \nu^{10} + 292392729702505742733160 \nu^{9} - 1040119147479516578145527 \nu^{8} + 388039704245625216000627 \nu^{7} + 506700705832317210409544 \nu^{6} + 4387546501364479788167939 \nu^{5} + 3561481521476851963965096 \nu^{4} - 36719641709862948496729891 \nu^{3} + 27844288339436115275679098 \nu^{2} + 49425586234727022770461941 \nu - 62351648937184846117813681$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{12}$$ $$=$$ $$($$$$-20975988167062581733 \nu^{15} + 39547953338415445145 \nu^{14} - 176449598889577838775 \nu^{13} + 274358583941958469564 \nu^{12} - 2164217419437062242990 \nu^{11} - 919597481073259125842 \nu^{10} + 1823706335460357174691 \nu^{9} - 20580179620448945951591 \nu^{8} + 8451228402253354843526 \nu^{7} - 93694473582422293608019 \nu^{6} + 5075887138800366711766 \nu^{5} - 35902260644185079638963 \nu^{4} - 523468395825219908972988 \nu^{3} + 334862658377387529702803 \nu^{2} - 868849722103095172193017 \nu - 916714740028087519831068$$$$)/$$$$24\!\cdots\!82$$ $$\beta_{13}$$ $$=$$ $$($$$$357782714812193812176 \nu^{15} - 123523917827887574891 \nu^{14} - 800114075158984515449 \nu^{13} + 1607794271396129055839 \nu^{12} + 17685313247604191574774 \nu^{11} + 86155282281315189826046 \nu^{10} - 231037252399480846221794 \nu^{9} - 157743783882519768425145 \nu^{8} + 810871220438588178709463 \nu^{7} + 566992698905865867472240 \nu^{6} + 1009287290081580835466493 \nu^{5} - 9850704470596476333646446 \nu^{4} + 1712964652569758738280733 \nu^{3} + 26803847027910109683168612 \nu^{2} - 22439135362612872985811991 \nu - 19147500634288685846910169$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{14}$$ $$=$$ $$($$$$476964966549267156748 \nu^{15} - 1473705395402246668575 \nu^{14} + 2999350746100514306003 \nu^{13} - 6596755340890207762998 \nu^{12} + 45185336926506497857202 \nu^{11} - 12469513850456023691368 \nu^{10} - 254095554940103528613762 \nu^{9} + 430971747898100629606361 \nu^{8} + 58620806557048961231735 \nu^{7} + 1194059572369888956687611 \nu^{6} - 2290539345841782308758671 \nu^{5} - 6240010475731289229690247 \nu^{4} + 16674572537573573808687019 \nu^{3} - 1448346950335563611661837 \nu^{2} - 18024602065451566144299597 \nu + 9148560862865248236217178$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{15}$$ $$=$$ $$($$$$570985375503210482467 \nu^{15} - 1520250623078270404265 \nu^{14} + 3751560263724227188644 \nu^{13} - 6673310293034086938068 \nu^{12} + 53067762013301206658762 \nu^{11} + 5500764870110710645582 \nu^{10} - 246766753353328108128263 \nu^{9} + 564892960196155782707651 \nu^{8} + 84055353158510742790295 \nu^{7} + 1438278968368169116260339 \nu^{6} - 1482438585718953558456509 \nu^{5} - 6069367767126172146850815 \nu^{4} + 21509477402988567555648733 \nu^{3} - 2777676281482766127449851 \nu^{2} - 20004927468191489674451150 \nu + 27664828216431640824705446$$$$)/$$$$41\!\cdots\!94$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{15} + \beta_{10} + 3 \beta_{8} - 3 \beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{15} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 3 \beta_{7} + 6 \beta_{6} - 11 \beta_{5} - 9 \beta_{4} - 2 \beta_{2} - 3 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{15} - 9 \beta_{14} - 2 \beta_{13} - 2 \beta_{9} - 27 \beta_{8} - 27 \beta_{7} + 13 \beta_{6} + 29 \beta_{5} - 19 \beta_{4} - 9 \beta_{3} - 13 \beta_{2} - 42$$ $$\nu^{5}$$ $$=$$ $$-4 \beta_{15} - 36 \beta_{14} + 13 \beta_{13} - 13 \beta_{12} - 9 \beta_{11} + 23 \beta_{10} + 9 \beta_{9} + 34 \beta_{8} - 10 \beta_{7} - 21 \beta_{6} + 61 \beta_{5} + 42 \beta_{4} - 44 \beta_{3} + 42 \beta_{2} - 30 \beta_{1} - 73$$ $$\nu^{6}$$ $$=$$ $$7 \beta_{15} + 7 \beta_{14} + 44 \beta_{13} - 76 \beta_{12} - 44 \beta_{11} - 142 \beta_{10} - 65 \beta_{8} + 209 \beta_{7} + 98 \beta_{3} + 105 \beta_{2} - 191 \beta_{1} + 209$$ $$\nu^{7}$$ $$=$$ $$449 \beta_{15} + 59 \beta_{14} - 59 \beta_{13} + 157 \beta_{12} + 98 \beta_{11} - 504 \beta_{10} + 59 \beta_{9} - 551 \beta_{8} + 489 \beta_{7} - 143 \beta_{6} + 710 \beta_{5} + 257 \beta_{4} + 390 \beta_{3} - 59 \beta_{2} + 316 \beta_{1} - 59$$ $$\nu^{8}$$ $$=$$ $$11 \beta_{15} + 378 \beta_{14} + 602 \beta_{12} + 390 \beta_{11} + 591 \beta_{10} + 602 \beta_{9} + 2340 \beta_{8} + 1312 \beta_{7} - 1589 \beta_{6} - 434 \beta_{5} + 2901 \beta_{4} + 710 \beta_{2} + 1312 \beta_{1} + 1028$$ $$\nu^{9}$$ $$=$$ $$-2504 \beta_{15} + 4164 \beta_{14} - 201 \beta_{13} - 367 \beta_{9} + 3660 \beta_{8} + 3660 \beta_{7} + 635 \beta_{6} - 12024 \beta_{5} - 1320 \beta_{4} + 4164 \beta_{3} - 635 \beta_{2} + 11389$$ $$\nu^{10}$$ $$=$$ $$2504 \beta_{15} + 4696 \beta_{14} - 6668 \beta_{13} + 6668 \beta_{12} + 4164 \beta_{11} + 1972 \beta_{10} - 4164 \beta_{9} - 15494 \beta_{8} - 18735 \beta_{7} + 9866 \beta_{6} - 4121 \beta_{5} - 18692 \beta_{4} + 3312 \beta_{3} - 18692 \beta_{2} + 14030 \beta_{1} - 11612$$ $$\nu^{11}$$ $$=$$ $$-22394 \beta_{15} - 22394 \beta_{14} - 3312 \beta_{13} + 5504 \beta_{12} + 3312 \beta_{11} + 67722 \beta_{10} + 42552 \beta_{8} - 76812 \beta_{7} - 64410 \beta_{3} - 7433 \beta_{2} + 12584 \beta_{1} - 76812$$ $$\nu^{12}$$ $$=$$ $$-102120 \beta_{15} - 39824 \beta_{14} + 39824 \beta_{13} - 104234 \beta_{12} - 64410 \beta_{11} + 35457 \beta_{10} - 39824 \beta_{9} - 2325 \beta_{8} - 69559 \beta_{7} + 93965 \beta_{6} - 109959 \beta_{5} - 176694 \beta_{4} - 62296 \beta_{3} + 39824 \beta_{2} - 216518 \beta_{1} + 39824$$ $$\nu^{13}$$ $$=$$ $$300533 \beta_{15} - 247371 \beta_{14} - 99867 \beta_{12} - 62296 \beta_{11} - 400400 \beta_{10} - 99867 \beta_{9} - 905140 \beta_{8} - 209826 \beta_{7} + 241814 \beta_{6} + 945646 \beta_{5} - 451640 \beta_{4} - 109959 \beta_{2} - 209826 \beta_{1} - 695314$$ $$\nu^{14}$$ $$=$$ $$594635 \beta_{15} - 966540 \beta_{14} + 338104 \beta_{13} + 547904 \beta_{9} + 786321 \beta_{8} + 786321 \beta_{7} - 1283750 \beta_{6} + 2729161 \beta_{5} + 2417009 \beta_{4} - 966540 \beta_{3} + 1283750 \beta_{2} - 1445411$$ $$\nu^{15}$$ $$=$$ $$-594635 \beta_{15} + 1832695 \beta_{14} + 1561175 \beta_{13} - 1561175 \beta_{12} - 966540 \beta_{11} - 3393870 \beta_{10} + 966540 \beta_{9} + 3573048 \beta_{8} + 9449472 \beta_{7} - 2278463 \beta_{6} - 4192596 \beta_{5} + 4290336 \beta_{4} + 3933458 \beta_{3} + 4290336 \beta_{2} - 3245003 \beta_{1} + 11063785$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$-\beta_{7}$$ $$1$$

Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 2.24350 − 2.23726i 1.60675 − 1.36085i −0.797732 + 1.94863i −1.43448 + 2.82504i −1.95510 − 0.109518i −1.29715 + 0.104262i 0.988132 + 0.846795i 1.64608 + 1.06057i 2.24350 + 2.23726i 1.60675 + 1.36085i −0.797732 − 1.94863i −1.43448 − 2.82504i −1.95510 + 0.109518i −1.29715 − 0.104262i 0.988132 − 0.846795i 1.64608 − 1.06057i
−0.577539 + 0.794915i −0.535233 + 1.64728i 0.937730 + 2.88604i −0.321645 + 0.233689i −1.00033 1.37683i 6.87311 2.23321i −6.57364 2.13591i −2.42705 1.76336i 0.390645i
7.2 −0.184008 + 0.253266i 0.535233 1.64728i 1.20578 + 3.71102i 5.99919 4.35866i 0.318712 + 0.438669i −9.53633 + 3.09854i −2.35267 0.764430i −2.42705 1.76336i 2.32142i
7.3 1.30204 1.79211i 0.535233 1.64728i −0.280267 0.862573i −7.03442 + 5.11081i −2.25520 3.10402i 6.34535 2.06173i 6.51625 + 2.11726i −2.42705 1.76336i 19.2609i
7.4 1.69557 2.33376i −0.535233 + 1.64728i −1.33538 4.10989i 0.356879 0.259287i 2.93682 + 4.04219i −10.0641 + 3.27002i −0.881730 0.286491i −2.42705 1.76336i 1.27251i
13.1 −3.47243 1.12826i −1.40126 + 1.01807i 7.54873 + 5.48447i 1.69033 + 5.20232i 6.01443 1.95421i −4.20886 + 5.79300i −11.4402 15.7461i 0.927051 2.85317i 19.9718i
13.2 −2.40785 0.782357i 1.40126 1.01807i 1.94958 + 1.41645i −2.61024 8.03348i −4.17052 + 1.35508i 1.43445 1.97435i 2.36641 + 3.25708i 0.927051 2.85317i 21.3855i
13.3 1.28981 + 0.419086i 1.40126 1.01807i −1.74808 1.27006i 0.708979 + 2.18201i 2.23402 0.725878i −5.74346 + 7.90520i −4.91103 6.75946i 0.927051 2.85317i 3.11151i
13.4 2.35440 + 0.764990i −1.40126 + 1.01807i 1.72190 + 1.25104i −0.789076 2.42853i −4.07793 + 1.32500i −0.100159 + 0.137856i −2.72337 3.74840i 0.927051 2.85317i 6.32135i
19.1 −0.577539 0.794915i −0.535233 1.64728i 0.937730 2.88604i −0.321645 0.233689i −1.00033 + 1.37683i 6.87311 + 2.23321i −6.57364 + 2.13591i −2.42705 + 1.76336i 0.390645i
19.2 −0.184008 0.253266i 0.535233 + 1.64728i 1.20578 3.71102i 5.99919 + 4.35866i 0.318712 0.438669i −9.53633 3.09854i −2.35267 + 0.764430i −2.42705 + 1.76336i 2.32142i
19.3 1.30204 + 1.79211i 0.535233 + 1.64728i −0.280267 + 0.862573i −7.03442 5.11081i −2.25520 + 3.10402i 6.34535 + 2.06173i 6.51625 2.11726i −2.42705 + 1.76336i 19.2609i
19.4 1.69557 + 2.33376i −0.535233 1.64728i −1.33538 + 4.10989i 0.356879 + 0.259287i 2.93682 4.04219i −10.0641 3.27002i −0.881730 + 0.286491i −2.42705 + 1.76336i 1.27251i
28.1 −3.47243 + 1.12826i −1.40126 1.01807i 7.54873 5.48447i 1.69033 5.20232i 6.01443 + 1.95421i −4.20886 5.79300i −11.4402 + 15.7461i 0.927051 + 2.85317i 19.9718i
28.2 −2.40785 + 0.782357i 1.40126 + 1.01807i 1.94958 1.41645i −2.61024 + 8.03348i −4.17052 1.35508i 1.43445 + 1.97435i 2.36641 3.25708i 0.927051 + 2.85317i 21.3855i
28.3 1.28981 0.419086i 1.40126 + 1.01807i −1.74808 + 1.27006i 0.708979 2.18201i 2.23402 + 0.725878i −5.74346 7.90520i −4.91103 + 6.75946i 0.927051 + 2.85317i 3.11151i
28.4 2.35440 0.764990i −1.40126 1.01807i 1.72190 1.25104i −0.789076 + 2.42853i −4.07793 1.32500i −0.100159 0.137856i −2.72337 + 3.74840i 0.927051 + 2.85317i 6.32135i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 28.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.3.g.a 16
3.b odd 2 1 99.3.k.c 16
4.b odd 2 1 528.3.bf.b 16
11.b odd 2 1 363.3.g.f 16
11.c even 5 1 363.3.c.e 16
11.c even 5 1 363.3.g.a 16
11.c even 5 1 363.3.g.f 16
11.c even 5 1 363.3.g.g 16
11.d odd 10 1 inner 33.3.g.a 16
11.d odd 10 1 363.3.c.e 16
11.d odd 10 1 363.3.g.a 16
11.d odd 10 1 363.3.g.g 16
33.f even 10 1 99.3.k.c 16
33.f even 10 1 1089.3.c.m 16
33.h odd 10 1 1089.3.c.m 16
44.g even 10 1 528.3.bf.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.g.a 16 1.a even 1 1 trivial
33.3.g.a 16 11.d odd 10 1 inner
99.3.k.c 16 3.b odd 2 1
99.3.k.c 16 33.f even 10 1
363.3.c.e 16 11.c even 5 1
363.3.c.e 16 11.d odd 10 1
363.3.g.a 16 11.c even 5 1
363.3.g.a 16 11.d odd 10 1
363.3.g.f 16 11.b odd 2 1
363.3.g.f 16 11.c even 5 1
363.3.g.g 16 11.c even 5 1
363.3.g.g 16 11.d odd 10 1
528.3.bf.b 16 4.b odd 2 1
528.3.bf.b 16 44.g even 10 1
1089.3.c.m 16 33.f even 10 1
1089.3.c.m 16 33.h odd 10 1

Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(33, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 8 T^{4} + 4 T^{6} - 190 T^{7} - 123 T^{8} + 330 T^{9} - 460 T^{10} - 670 T^{11} + 107 T^{12} - 3270 T^{13} + 14283 T^{14} + 30390 T^{15} - 41575 T^{16} + 121560 T^{17} + 228528 T^{18} - 209280 T^{19} + 27392 T^{20} - 686080 T^{21} - 1884160 T^{22} + 5406720 T^{23} - 8060928 T^{24} - 49807360 T^{25} + 4194304 T^{26} + 134217728 T^{28} - 536870912 T^{30} + 4294967296 T^{32}$$
$3$ $$( 1 + 3 T^{2} + 9 T^{4} + 27 T^{6} + 81 T^{8} )^{2}$$
$5$ $$1 + 4 T - 41 T^{2} - 326 T^{3} + 556 T^{4} + 8402 T^{5} - 24744 T^{6} - 158358 T^{7} + 1206372 T^{8} + 5683018 T^{9} - 36533679 T^{10} - 90990212 T^{11} + 1135973351 T^{12} + 223262516 T^{13} - 39368366016 T^{14} - 2447632564 T^{15} + 1162197815296 T^{16} - 61190814100 T^{17} - 24605228760000 T^{18} + 3488476812500 T^{19} + 443739590234375 T^{20} - 888576289062500 T^{21} - 8919355224609375 T^{22} + 34686389160156250 T^{23} + 184077758789062500 T^{24} - 604087829589843750 T^{25} - 2359771728515625000 T^{26} + 20031929016113281250 T^{27} + 33140182495117187500 T^{28} -$$$$48\!\cdots\!50$$$$T^{29} -$$$$15\!\cdots\!25$$$$T^{30} +$$$$37\!\cdots\!00$$$$T^{31} +$$$$23\!\cdots\!25$$$$T^{32}$$
$7$ $$1 + 30 T + 443 T^{2} + 5050 T^{3} + 56021 T^{4} + 586310 T^{5} + 5549342 T^{6} + 50952170 T^{7} + 468313746 T^{8} + 4074185190 T^{9} + 33114120719 T^{10} + 264330080230 T^{11} + 2092010733594 T^{12} + 15864828691680 T^{13} + 116529967249246 T^{14} + 855941511341600 T^{15} + 6146534444128017 T^{16} + 41941134055738400 T^{17} + 279788451365439646 T^{18} + 1866481230747460320 T^{19} + 12060025569033424794 T^{20} + 74666705231159227270 T^{21} +$$$$45\!\cdots\!19$$$$T^{22} +$$$$27\!\cdots\!10$$$$T^{23} +$$$$15\!\cdots\!46$$$$T^{24} +$$$$82\!\cdots\!30$$$$T^{25} +$$$$44\!\cdots\!42$$$$T^{26} +$$$$22\!\cdots\!90$$$$T^{27} +$$$$10\!\cdots\!21$$$$T^{28} +$$$$47\!\cdots\!50$$$$T^{29} +$$$$20\!\cdots\!43$$$$T^{30} +$$$$67\!\cdots\!70$$$$T^{31} +$$$$11\!\cdots\!01$$$$T^{32}$$
$11$ $$1 + 10 T + 125 T^{2} + 530 T^{3} - 501 T^{4} - 145970 T^{5} + 1434455 T^{6} + 24556950 T^{7} + 379728976 T^{8} + 2971390950 T^{9} + 21001855655 T^{10} - 258594759170 T^{11} - 107393799381 T^{12} + 13746835038530 T^{13} + 392303547090125 T^{14} + 3797498335832410 T^{15} + 45949729863572161 T^{16}$$
$13$ $$1 - 30 T + 587 T^{2} - 11750 T^{3} + 232289 T^{4} - 3942850 T^{5} + 60373070 T^{6} - 849756370 T^{7} + 11683550898 T^{8} - 148143792450 T^{9} + 1704050844755 T^{10} - 17335247746130 T^{11} + 149383368485646 T^{12} - 1122414950958600 T^{13} + 6515222601436258 T^{14} + 31611597343900820 T^{15} - 1160324628036481527 T^{16} + 5342359951119238580 T^{17} +$$$$18\!\cdots\!38$$$$T^{18} -$$$$54\!\cdots\!00$$$$T^{19} +$$$$12\!\cdots\!66$$$$T^{20} -$$$$23\!\cdots\!70$$$$T^{21} +$$$$39\!\cdots\!55$$$$T^{22} -$$$$58\!\cdots\!50$$$$T^{23} +$$$$77\!\cdots\!18$$$$T^{24} -$$$$95\!\cdots\!30$$$$T^{25} +$$$$11\!\cdots\!70$$$$T^{26} -$$$$12\!\cdots\!50$$$$T^{27} +$$$$12\!\cdots\!29$$$$T^{28} -$$$$10\!\cdots\!50$$$$T^{29} +$$$$91\!\cdots\!27$$$$T^{30} -$$$$78\!\cdots\!70$$$$T^{31} +$$$$44\!\cdots\!81$$$$T^{32}$$
$17$ $$1 + 10 T + 820 T^{2} + 21310 T^{3} + 390646 T^{4} + 13325660 T^{5} + 198026940 T^{6} + 4017398430 T^{7} + 78579088635 T^{8} + 650290141900 T^{9} + 10925791113180 T^{10} + 18747432168760 T^{11} - 4542291295151416 T^{12} - 85648097191686310 T^{13} - 2616538510506970920 T^{14} - 56626351206035192470 T^{15} -$$$$81\!\cdots\!11$$$$T^{16} -$$$$16\!\cdots\!30$$$$T^{17} -$$$$21\!\cdots\!20$$$$T^{18} -$$$$20\!\cdots\!90$$$$T^{19} -$$$$31\!\cdots\!56$$$$T^{20} +$$$$37\!\cdots\!40$$$$T^{21} +$$$$63\!\cdots\!80$$$$T^{22} +$$$$10\!\cdots\!00$$$$T^{23} +$$$$38\!\cdots\!35$$$$T^{24} +$$$$56\!\cdots\!70$$$$T^{25} +$$$$80\!\cdots\!40$$$$T^{26} +$$$$15\!\cdots\!40$$$$T^{27} +$$$$13\!\cdots\!66$$$$T^{28} +$$$$20\!\cdots\!90$$$$T^{29} +$$$$23\!\cdots\!20$$$$T^{30} +$$$$81\!\cdots\!90$$$$T^{31} +$$$$23\!\cdots\!61$$$$T^{32}$$
$19$ $$1 + 68 T^{2} - 24500 T^{3} + 194954 T^{4} - 1666000 T^{5} + 288026060 T^{6} - 3913548340 T^{7} + 46342771203 T^{8} - 2296827284640 T^{9} + 48518556742580 T^{10} - 543301305394100 T^{11} + 17817435718051236 T^{12} - 505288210780128180 T^{13} + 4817696834455760272 T^{14} -$$$$11\!\cdots\!20$$$$T^{15} +$$$$42\!\cdots\!73$$$$T^{16} -$$$$40\!\cdots\!20$$$$T^{17} +$$$$62\!\cdots\!12$$$$T^{18} -$$$$23\!\cdots\!80$$$$T^{19} +$$$$30\!\cdots\!76$$$$T^{20} -$$$$33\!\cdots\!00$$$$T^{21} +$$$$10\!\cdots\!80$$$$T^{22} -$$$$18\!\cdots\!40$$$$T^{23} +$$$$13\!\cdots\!43$$$$T^{24} -$$$$40\!\cdots\!40$$$$T^{25} +$$$$10\!\cdots\!60$$$$T^{26} -$$$$22\!\cdots\!00$$$$T^{27} +$$$$95\!\cdots\!34$$$$T^{28} -$$$$43\!\cdots\!00$$$$T^{29} +$$$$43\!\cdots\!88$$$$T^{30} +$$$$83\!\cdots\!61$$$$T^{32}$$
$23$ $$( 1 - 66 T + 5094 T^{2} - 230356 T^{3} + 10182141 T^{4} - 348333808 T^{5} + 11089711126 T^{6} - 298981380378 T^{7} + 7385976715852 T^{8} - 158161150219962 T^{9} + 3103355851210966 T^{10} - 51565904936035312 T^{11} + 797373493980066621 T^{12} - 9542845417131329044 T^{13} +$$$$11\!\cdots\!74$$$$T^{14} -$$$$76\!\cdots\!94$$$$T^{15} +$$$$61\!\cdots\!61$$$$T^{16} )^{2}$$
$29$ $$1 - 160 T + 15013 T^{2} - 990140 T^{3} + 49719364 T^{4} - 1931787020 T^{5} + 55026326860 T^{6} - 840259706100 T^{7} - 19139100675292 T^{8} + 2108640129773860 T^{9} - 96767641133217765 T^{10} + 3027695154885413600 T^{11} - 62310681933819745809 T^{12} +$$$$27\!\cdots\!60$$$$T^{13} +$$$$45\!\cdots\!32$$$$T^{14} -$$$$25\!\cdots\!40$$$$T^{15} +$$$$88\!\cdots\!68$$$$T^{16} -$$$$21\!\cdots\!40$$$$T^{17} +$$$$32\!\cdots\!92$$$$T^{18} +$$$$16\!\cdots\!60$$$$T^{19} -$$$$31\!\cdots\!49$$$$T^{20} +$$$$12\!\cdots\!00$$$$T^{21} -$$$$34\!\cdots\!65$$$$T^{22} +$$$$62\!\cdots\!60$$$$T^{23} -$$$$47\!\cdots\!32$$$$T^{24} -$$$$17\!\cdots\!00$$$$T^{25} +$$$$97\!\cdots\!60$$$$T^{26} -$$$$28\!\cdots\!20$$$$T^{27} +$$$$62\!\cdots\!84$$$$T^{28} -$$$$10\!\cdots\!40$$$$T^{29} +$$$$13\!\cdots\!93$$$$T^{30} -$$$$11\!\cdots\!60$$$$T^{31} +$$$$62\!\cdots\!41$$$$T^{32}$$
$31$ $$1 - 10 T - 2495 T^{2} - 50440 T^{3} + 4737748 T^{4} + 140647810 T^{5} - 3464845740 T^{6} - 277335042990 T^{7} + 1019957604708 T^{8} + 293544248236760 T^{9} + 4654315812995055 T^{10} - 311592266592737110 T^{11} - 7772692308833591089 T^{12} +$$$$19\!\cdots\!00$$$$T^{13} +$$$$12\!\cdots\!00$$$$T^{14} -$$$$10\!\cdots\!00$$$$T^{15} -$$$$12\!\cdots\!00$$$$T^{16} -$$$$96\!\cdots\!00$$$$T^{17} +$$$$11\!\cdots\!00$$$$T^{18} +$$$$17\!\cdots\!00$$$$T^{19} -$$$$66\!\cdots\!49$$$$T^{20} -$$$$25\!\cdots\!10$$$$T^{21} +$$$$36\!\cdots\!55$$$$T^{22} +$$$$22\!\cdots\!60$$$$T^{23} +$$$$74\!\cdots\!48$$$$T^{24} -$$$$19\!\cdots\!90$$$$T^{25} -$$$$23\!\cdots\!40$$$$T^{26} +$$$$90\!\cdots\!10$$$$T^{27} +$$$$29\!\cdots\!08$$$$T^{28} -$$$$30\!\cdots\!40$$$$T^{29} -$$$$14\!\cdots\!95$$$$T^{30} -$$$$55\!\cdots\!10$$$$T^{31} +$$$$52\!\cdots\!61$$$$T^{32}$$
$37$ $$1 + 126 T + 1723 T^{2} - 412814 T^{3} - 18214575 T^{4} + 351923510 T^{5} + 43093576938 T^{6} + 836591530018 T^{7} - 32958031243238 T^{8} - 2093615440151926 T^{9} - 19792102407878669 T^{10} + 1590648037174305038 T^{11} + 62026368402590720226 T^{12} +$$$$49\!\cdots\!72$$$$T^{13} -$$$$40\!\cdots\!62$$$$T^{14} -$$$$78\!\cdots\!60$$$$T^{15} +$$$$21\!\cdots\!13$$$$T^{16} -$$$$10\!\cdots\!40$$$$T^{17} -$$$$75\!\cdots\!82$$$$T^{18} +$$$$12\!\cdots\!48$$$$T^{19} +$$$$21\!\cdots\!46$$$$T^{20} +$$$$76\!\cdots\!62$$$$T^{21} -$$$$13\!\cdots\!89$$$$T^{22} -$$$$18\!\cdots\!14$$$$T^{23} -$$$$40\!\cdots\!58$$$$T^{24} +$$$$14\!\cdots\!22$$$$T^{25} +$$$$99\!\cdots\!38$$$$T^{26} +$$$$11\!\cdots\!90$$$$T^{27} -$$$$78\!\cdots\!75$$$$T^{28} -$$$$24\!\cdots\!26$$$$T^{29} +$$$$13\!\cdots\!83$$$$T^{30} +$$$$14\!\cdots\!74$$$$T^{31} +$$$$15\!\cdots\!81$$$$T^{32}$$
$41$ $$1 + 120 T + 11625 T^{2} + 945480 T^{3} + 63040941 T^{4} + 3835690920 T^{5} + 205787619310 T^{6} + 10127082489930 T^{7} + 464765909051670 T^{8} + 19526181575237790 T^{9} + 774755421171593245 T^{10} + 28972958068309133550 T^{11} +$$$$10\!\cdots\!74$$$$T^{12} +$$$$34\!\cdots\!70$$$$T^{13} +$$$$11\!\cdots\!10$$$$T^{14} +$$$$37\!\cdots\!70$$$$T^{15} +$$$$14\!\cdots\!49$$$$T^{16} +$$$$63\!\cdots\!70$$$$T^{17} +$$$$31\!\cdots\!10$$$$T^{18} +$$$$16\!\cdots\!70$$$$T^{19} +$$$$80\!\cdots\!54$$$$T^{20} +$$$$38\!\cdots\!50$$$$T^{21} +$$$$17\!\cdots\!45$$$$T^{22} +$$$$74\!\cdots\!90$$$$T^{23} +$$$$29\!\cdots\!70$$$$T^{24} +$$$$10\!\cdots\!30$$$$T^{25} +$$$$37\!\cdots\!10$$$$T^{26} +$$$$11\!\cdots\!20$$$$T^{27} +$$$$32\!\cdots\!01$$$$T^{28} +$$$$80\!\cdots\!80$$$$T^{29} +$$$$16\!\cdots\!25$$$$T^{30} +$$$$29\!\cdots\!20$$$$T^{31} +$$$$40\!\cdots\!81$$$$T^{32}$$
$43$ $$1 - 12876 T^{2} + 91501662 T^{4} - 458248399064 T^{6} + 1782172921950441 T^{8} - 5666821055429722520 T^{10} +$$$$15\!\cdots\!78$$$$T^{12} -$$$$34\!\cdots\!80$$$$T^{14} +$$$$69\!\cdots\!12$$$$T^{16} -$$$$11\!\cdots\!80$$$$T^{18} +$$$$17\!\cdots\!78$$$$T^{20} -$$$$22\!\cdots\!20$$$$T^{22} +$$$$24\!\cdots\!41$$$$T^{24} -$$$$21\!\cdots\!64$$$$T^{26} +$$$$14\!\cdots\!62$$$$T^{28} -$$$$70\!\cdots\!76$$$$T^{30} +$$$$18\!\cdots\!01$$$$T^{32}$$
$47$ $$1 + 150 T + 5703 T^{2} - 111290 T^{3} + 1189236 T^{4} + 1478901940 T^{5} + 65812406892 T^{6} - 349108466940 T^{7} - 11262862832804 T^{8} + 7417285475912590 T^{9} + 389924112150834429 T^{10} + 6636949241471966250 T^{11} +$$$$25\!\cdots\!19$$$$T^{12} +$$$$23\!\cdots\!60$$$$T^{13} +$$$$10\!\cdots\!16$$$$T^{14} +$$$$44\!\cdots\!40$$$$T^{15} +$$$$22\!\cdots\!12$$$$T^{16} +$$$$97\!\cdots\!60$$$$T^{17} +$$$$51\!\cdots\!96$$$$T^{18} +$$$$25\!\cdots\!40$$$$T^{19} +$$$$61\!\cdots\!59$$$$T^{20} +$$$$34\!\cdots\!50$$$$T^{21} +$$$$45\!\cdots\!89$$$$T^{22} +$$$$19\!\cdots\!10$$$$T^{23} -$$$$63\!\cdots\!84$$$$T^{24} -$$$$43\!\cdots\!60$$$$T^{25} +$$$$18\!\cdots\!92$$$$T^{26} +$$$$90\!\cdots\!60$$$$T^{27} +$$$$16\!\cdots\!16$$$$T^{28} -$$$$33\!\cdots\!10$$$$T^{29} +$$$$37\!\cdots\!83$$$$T^{30} +$$$$21\!\cdots\!50$$$$T^{31} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$1 - 342 T + 56425 T^{2} - 5825308 T^{3} + 395226372 T^{4} - 14794732502 T^{5} - 224283930792 T^{6} + 78367975406558 T^{7} - 6103771119756596 T^{8} + 258849959496378940 T^{9} - 2857462886249188349 T^{10} -$$$$46\!\cdots\!66$$$$T^{11} +$$$$40\!\cdots\!59$$$$T^{12} -$$$$17\!\cdots\!00$$$$T^{13} +$$$$30\!\cdots\!16$$$$T^{14} +$$$$15\!\cdots\!92$$$$T^{15} -$$$$14\!\cdots\!56$$$$T^{16} +$$$$42\!\cdots\!28$$$$T^{17} +$$$$24\!\cdots\!96$$$$T^{18} -$$$$38\!\cdots\!00$$$$T^{19} +$$$$25\!\cdots\!99$$$$T^{20} -$$$$81\!\cdots\!34$$$$T^{21} -$$$$14\!\cdots\!09$$$$T^{22} +$$$$35\!\cdots\!60$$$$T^{23} -$$$$23\!\cdots\!16$$$$T^{24} +$$$$85\!\cdots\!62$$$$T^{25} -$$$$68\!\cdots\!92$$$$T^{26} -$$$$12\!\cdots\!18$$$$T^{27} +$$$$95\!\cdots\!32$$$$T^{28} -$$$$39\!\cdots\!32$$$$T^{29} +$$$$10\!\cdots\!25$$$$T^{30} -$$$$18\!\cdots\!58$$$$T^{31} +$$$$15\!\cdots\!41$$$$T^{32}$$
$59$ $$1 - 110 T - 1820 T^{2} + 874210 T^{3} - 21932762 T^{4} - 3787520260 T^{5} + 169192636440 T^{6} + 16984414752090 T^{7} - 1089925211772957 T^{8} - 63720166969996940 T^{9} + 5917794901333836780 T^{10} +$$$$16\!\cdots\!40$$$$T^{11} -$$$$24\!\cdots\!44$$$$T^{12} -$$$$47\!\cdots\!50$$$$T^{13} +$$$$10\!\cdots\!00$$$$T^{14} +$$$$83\!\cdots\!50$$$$T^{15} -$$$$39\!\cdots\!75$$$$T^{16} +$$$$29\!\cdots\!50$$$$T^{17} +$$$$12\!\cdots\!00$$$$T^{18} -$$$$20\!\cdots\!50$$$$T^{19} -$$$$35\!\cdots\!24$$$$T^{20} +$$$$84\!\cdots\!40$$$$T^{21} +$$$$10\!\cdots\!80$$$$T^{22} -$$$$39\!\cdots\!40$$$$T^{23} -$$$$23\!\cdots\!37$$$$T^{24} +$$$$12\!\cdots\!90$$$$T^{25} +$$$$44\!\cdots\!40$$$$T^{26} -$$$$34\!\cdots\!60$$$$T^{27} -$$$$69\!\cdots\!82$$$$T^{28} +$$$$96\!\cdots\!10$$$$T^{29} -$$$$69\!\cdots\!20$$$$T^{30} -$$$$14\!\cdots\!10$$$$T^{31} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$1 + 90 T + 5185 T^{2} + 111300 T^{3} + 5561708 T^{4} + 355418970 T^{5} + 44628720040 T^{6} + 1586771608090 T^{7} - 2104031885352 T^{8} - 10404552243962580 T^{9} - 1081281936652696465 T^{10} - 65423743873501326530 T^{11} -$$$$19\!\cdots\!49$$$$T^{12} +$$$$44\!\cdots\!80$$$$T^{13} -$$$$23\!\cdots\!00$$$$T^{14} -$$$$53\!\cdots\!80$$$$T^{15} -$$$$50\!\cdots\!80$$$$T^{16} -$$$$20\!\cdots\!80$$$$T^{17} -$$$$31\!\cdots\!00$$$$T^{18} +$$$$22\!\cdots\!80$$$$T^{19} -$$$$37\!\cdots\!69$$$$T^{20} -$$$$46\!\cdots\!30$$$$T^{21} -$$$$28\!\cdots\!65$$$$T^{22} -$$$$10\!\cdots\!80$$$$T^{23} -$$$$77\!\cdots\!72$$$$T^{24} +$$$$21\!\cdots\!90$$$$T^{25} +$$$$22\!\cdots\!40$$$$T^{26} +$$$$67\!\cdots\!70$$$$T^{27} +$$$$39\!\cdots\!28$$$$T^{28} +$$$$29\!\cdots\!00$$$$T^{29} +$$$$50\!\cdots\!85$$$$T^{30} +$$$$32\!\cdots\!90$$$$T^{31} +$$$$13\!\cdots\!21$$$$T^{32}$$
$67$ $$( 1 - 18 T + 21975 T^{2} - 668180 T^{3} + 239231415 T^{4} - 8253401324 T^{5} + 1771382320897 T^{6} - 55179990303630 T^{7} + 9425465643780880 T^{8} - 247702976472995070 T^{9} + 35695339485656275537 T^{10} -$$$$74\!\cdots\!56$$$$T^{11} +$$$$97\!\cdots\!15$$$$T^{12} -$$$$12\!\cdots\!20$$$$T^{13} +$$$$17\!\cdots\!75$$$$T^{14} -$$$$66\!\cdots\!22$$$$T^{15} +$$$$16\!\cdots\!81$$$$T^{16} )^{2}$$
$71$ $$1 + 236 T + 10027 T^{2} - 2371506 T^{3} - 303058608 T^{4} - 188446842 T^{5} + 2379677829156 T^{6} + 145240013200230 T^{7} - 6446716975029360 T^{8} - 1111666934803343322 T^{9} - 20084120671257805311 T^{10} +$$$$40\!\cdots\!44$$$$T^{11} +$$$$23\!\cdots\!83$$$$T^{12} -$$$$82\!\cdots\!76$$$$T^{13} -$$$$11\!\cdots\!12$$$$T^{14} +$$$$77\!\cdots\!36$$$$T^{15} +$$$$49\!\cdots\!64$$$$T^{16} +$$$$39\!\cdots\!76$$$$T^{17} -$$$$28\!\cdots\!72$$$$T^{18} -$$$$10\!\cdots\!96$$$$T^{19} +$$$$14\!\cdots\!63$$$$T^{20} +$$$$13\!\cdots\!44$$$$T^{21} -$$$$32\!\cdots\!51$$$$T^{22} -$$$$91\!\cdots\!82$$$$T^{23} -$$$$26\!\cdots\!60$$$$T^{24} +$$$$30\!\cdots\!30$$$$T^{25} +$$$$25\!\cdots\!56$$$$T^{26} -$$$$10\!\cdots\!22$$$$T^{27} -$$$$81\!\cdots\!48$$$$T^{28} -$$$$32\!\cdots\!26$$$$T^{29} +$$$$68\!\cdots\!47$$$$T^{30} +$$$$81\!\cdots\!36$$$$T^{31} +$$$$17\!\cdots\!41$$$$T^{32}$$
$73$ $$1 + 350 T + 71113 T^{2} + 10598660 T^{3} + 1285017928 T^{4} + 133684739230 T^{5} + 12256794539364 T^{6} + 1004390235106950 T^{7} + 73846073626981752 T^{8} + 4829859050716130540 T^{9} +$$$$26\!\cdots\!55$$$$T^{10} +$$$$10\!\cdots\!50$$$$T^{11} +$$$$86\!\cdots\!87$$$$T^{12} -$$$$60\!\cdots\!80$$$$T^{13} -$$$$85\!\cdots\!12$$$$T^{14} -$$$$83\!\cdots\!20$$$$T^{15} -$$$$66\!\cdots\!40$$$$T^{16} -$$$$44\!\cdots\!80$$$$T^{17} -$$$$24\!\cdots\!92$$$$T^{18} -$$$$92\!\cdots\!20$$$$T^{19} +$$$$70\!\cdots\!47$$$$T^{20} +$$$$46\!\cdots\!50$$$$T^{21} +$$$$61\!\cdots\!55$$$$T^{22} +$$$$58\!\cdots\!60$$$$T^{23} +$$$$48\!\cdots\!72$$$$T^{24} +$$$$34\!\cdots\!50$$$$T^{25} +$$$$22\!\cdots\!64$$$$T^{26} +$$$$13\!\cdots\!70$$$$T^{27} +$$$$67\!\cdots\!48$$$$T^{28} +$$$$29\!\cdots\!40$$$$T^{29} +$$$$10\!\cdots\!53$$$$T^{30} +$$$$27\!\cdots\!50$$$$T^{31} +$$$$42\!\cdots\!21$$$$T^{32}$$
$79$ $$1 - 210 T + 33391 T^{2} - 5683230 T^{3} + 793281737 T^{4} - 99138120690 T^{5} + 12095814452554 T^{6} - 1372834116707750 T^{7} + 146848726194339006 T^{8} - 15133095124360319250 T^{9} +$$$$14\!\cdots\!75$$$$T^{10} -$$$$14\!\cdots\!50$$$$T^{11} +$$$$12\!\cdots\!18$$$$T^{12} -$$$$11\!\cdots\!40$$$$T^{13} +$$$$98\!\cdots\!50$$$$T^{14} -$$$$81\!\cdots\!40$$$$T^{15} +$$$$65\!\cdots\!97$$$$T^{16} -$$$$50\!\cdots\!40$$$$T^{17} +$$$$38\!\cdots\!50$$$$T^{18} -$$$$27\!\cdots\!40$$$$T^{19} +$$$$19\!\cdots\!98$$$$T^{20} -$$$$13\!\cdots\!50$$$$T^{21} +$$$$88\!\cdots\!75$$$$T^{22} -$$$$55\!\cdots\!50$$$$T^{23} +$$$$33\!\cdots\!26$$$$T^{24} -$$$$19\!\cdots\!50$$$$T^{25} +$$$$10\!\cdots\!54$$$$T^{26} -$$$$55\!\cdots\!90$$$$T^{27} +$$$$27\!\cdots\!97$$$$T^{28} -$$$$12\!\cdots\!30$$$$T^{29} +$$$$45\!\cdots\!51$$$$T^{30} -$$$$17\!\cdots\!10$$$$T^{31} +$$$$52\!\cdots\!41$$$$T^{32}$$
$83$ $$1 + 190 T + 67177 T^{2} + 11048590 T^{3} + 2116773229 T^{4} + 299366635430 T^{5} + 41314892504790 T^{6} + 4967390563524780 T^{7} + 550864440609675318 T^{8} + 55772626453864581280 T^{9} +$$$$52\!\cdots\!25$$$$T^{10} +$$$$44\!\cdots\!60$$$$T^{11} +$$$$35\!\cdots\!06$$$$T^{12} +$$$$26\!\cdots\!50$$$$T^{13} +$$$$18\!\cdots\!58$$$$T^{14} +$$$$13\!\cdots\!30$$$$T^{15} +$$$$10\!\cdots\!33$$$$T^{16} +$$$$94\!\cdots\!70$$$$T^{17} +$$$$89\!\cdots\!18$$$$T^{18} +$$$$85\!\cdots\!50$$$$T^{19} +$$$$80\!\cdots\!46$$$$T^{20} +$$$$68\!\cdots\!40$$$$T^{21} +$$$$55\!\cdots\!25$$$$T^{22} +$$$$41\!\cdots\!20$$$$T^{23} +$$$$27\!\cdots\!58$$$$T^{24} +$$$$17\!\cdots\!20$$$$T^{25} +$$$$99\!\cdots\!90$$$$T^{26} +$$$$49\!\cdots\!70$$$$T^{27} +$$$$24\!\cdots\!09$$$$T^{28} +$$$$86\!\cdots\!10$$$$T^{29} +$$$$36\!\cdots\!57$$$$T^{30} +$$$$70\!\cdots\!10$$$$T^{31} +$$$$25\!\cdots\!61$$$$T^{32}$$
$89$ $$( 1 - 38 T + 37118 T^{2} - 1566244 T^{3} + 705775677 T^{4} - 28974031040 T^{5} + 8919242373542 T^{6} - 334883278291638 T^{7} + 81740346306794380 T^{8} - 2652610447348064598 T^{9} +$$$$55\!\cdots\!22$$$$T^{10} -$$$$14\!\cdots\!40$$$$T^{11} +$$$$27\!\cdots\!37$$$$T^{12} -$$$$48\!\cdots\!44$$$$T^{13} +$$$$91\!\cdots\!78$$$$T^{14} -$$$$74\!\cdots\!58$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16} )^{2}$$
$97$ $$1 + 354 T + 19173 T^{2} - 6545856 T^{3} - 673044900 T^{4} + 77524200990 T^{5} + 12627807964588 T^{6} - 188800913648178 T^{7} - 100847199348278148 T^{8} - 1111689883210016304 T^{9} +$$$$55\!\cdots\!71$$$$T^{10} -$$$$94\!\cdots\!38$$$$T^{11} -$$$$58\!\cdots\!69$$$$T^{12} +$$$$47\!\cdots\!48$$$$T^{13} +$$$$12\!\cdots\!68$$$$T^{14} -$$$$29\!\cdots\!40$$$$T^{15} -$$$$16\!\cdots\!52$$$$T^{16} -$$$$27\!\cdots\!60$$$$T^{17} +$$$$10\!\cdots\!08$$$$T^{18} +$$$$39\!\cdots\!92$$$$T^{19} -$$$$45\!\cdots\!09$$$$T^{20} -$$$$69\!\cdots\!62$$$$T^{21} +$$$$38\!\cdots\!11$$$$T^{22} -$$$$72\!\cdots\!76$$$$T^{23} -$$$$61\!\cdots\!08$$$$T^{24} -$$$$10\!\cdots\!42$$$$T^{25} +$$$$68\!\cdots\!88$$$$T^{26} +$$$$39\!\cdots\!10$$$$T^{27} -$$$$32\!\cdots\!00$$$$T^{28} -$$$$29\!\cdots\!24$$$$T^{29} +$$$$81\!\cdots\!53$$$$T^{30} +$$$$14\!\cdots\!46$$$$T^{31} +$$$$37\!\cdots\!41$$$$T^{32}$$