Properties

 Label 1849.4.a.f Level 1849 Weight 4 Character orbit 1849.a Self dual yes Analytic conductor 109.095 Analytic rank 1 Dimension 10 CM no Inner twists 1

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1849 = 43^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1849.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$109.094531601$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - x^{9} - 59 x^{8} + 42 x^{7} + 1187 x^{6} - 541 x^{5} - 9389 x^{4} + 2180 x^{3} + 22676 x^{2} - 320 x - 768$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 43) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{5} q^{3} + ( 4 + \beta_{2} ) q^{4} + ( -2 + \beta_{7} ) q^{5} + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{6} + ( -5 - \beta_{2} - \beta_{6} ) q^{7} + ( 4 + 3 \beta_{1} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{8} + ( 12 + \beta_{1} - \beta_{4} - \beta_{7} + \beta_{8} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{5} q^{3} + ( 4 + \beta_{2} ) q^{4} + ( -2 + \beta_{7} ) q^{5} + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{6} + ( -5 - \beta_{2} - \beta_{6} ) q^{7} + ( 4 + 3 \beta_{1} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{8} + ( 12 + \beta_{1} - \beta_{4} - \beta_{7} + \beta_{8} ) q^{9} + ( -3 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} ) q^{10} + ( 3 - 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{11} + ( -6 - \beta_{1} - 4 \beta_{2} - \beta_{4} + 4 \beta_{5} - \beta_{8} + \beta_{9} ) q^{12} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{13} + ( -10 - 10 \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{14} + ( -6 + 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{15} + ( 5 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - 5 \beta_{5} + \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{16} + ( 6 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - \beta_{8} ) q^{17} + ( 26 + 14 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{18} + ( 8 + 8 \beta_{1} - \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{19} + ( -50 - 9 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{20} + ( -7 - 2 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} + 4 \beta_{6} + 5 \beta_{7} - 3 \beta_{8} ) q^{21} + ( -19 + 8 \beta_{1} + 4 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{22} + ( 2 - \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - 9 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} ) q^{23} + ( -18 - 23 \beta_{1} + 9 \beta_{5} + 7 \beta_{6} + 8 \beta_{7} - 6 \beta_{8} + \beta_{9} ) q^{24} + ( 12 + 3 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{25} + ( 7 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - \beta_{6} - 5 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{26} + ( 10 - \beta_{1} + 4 \beta_{2} + \beta_{3} + 4 \beta_{5} + 5 \beta_{6} - \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{27} + ( -80 + 12 \beta_{1} - 15 \beta_{2} - 4 \beta_{3} + \beta_{4} + 6 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{28} + ( -7 - 11 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 5 \beta_{6} - 6 \beta_{8} + \beta_{9} ) q^{29} + ( 65 + 5 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - 4 \beta_{6} - 14 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} ) q^{30} + ( -28 + 16 \beta_{1} + \beta_{2} - 4 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{31} + ( 37 - 5 \beta_{1} + 8 \beta_{2} - 7 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{32} + ( -45 - 7 \beta_{1} - 7 \beta_{2} - 10 \beta_{3} + 3 \beta_{5} + 9 \beta_{6} + 9 \beta_{7} - 6 \beta_{8} - \beta_{9} ) q^{33} + ( -35 + 16 \beta_{1} + 5 \beta_{2} + \beta_{3} + 5 \beta_{4} - 19 \beta_{5} - \beta_{6} + 11 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{34} + ( 43 - \beta_{1} + 8 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 8 \beta_{5} - 5 \beta_{6} - 8 \beta_{7} + 2 \beta_{8} ) q^{35} + ( 112 + 39 \beta_{1} + 13 \beta_{2} + \beta_{4} + 6 \beta_{5} + 3 \beta_{6} - 13 \beta_{7} + 4 \beta_{8} + \beta_{9} ) q^{36} + ( -6 + 18 \beta_{1} + 19 \beta_{2} + \beta_{3} + 5 \beta_{4} + 14 \beta_{5} + 14 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{37} + ( 87 - 10 \beta_{1} + 15 \beta_{2} + 7 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} + 12 \beta_{6} - 4 \beta_{7} - \beta_{8} - \beta_{9} ) q^{38} + ( -70 + 28 \beta_{1} - 8 \beta_{2} + 7 \beta_{3} + 2 \beta_{4} - 7 \beta_{6} + 2 \beta_{7} + 5 \beta_{8} ) q^{39} + ( -121 - 50 \beta_{1} - 11 \beta_{2} + 7 \beta_{3} + 2 \beta_{4} + 23 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} ) q^{40} + ( 28 - 28 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} - 23 \beta_{5} - \beta_{6} + 10 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} ) q^{41} + ( -24 + 34 \beta_{1} - 21 \beta_{2} - 14 \beta_{3} - 4 \beta_{4} - 41 \beta_{5} - 8 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} - \beta_{9} ) q^{42} + ( 18 + \beta_{1} + 11 \beta_{2} - 8 \beta_{3} + \beta_{4} - 36 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} - 7 \beta_{8} - \beta_{9} ) q^{44} + ( -110 - 20 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 18 \beta_{5} + 8 \beta_{6} + 6 \beta_{7} - 6 \beta_{8} ) q^{45} + ( -13 - 24 \beta_{1} + 7 \beta_{2} - 9 \beta_{3} - 21 \beta_{5} + 9 \beta_{6} + 3 \beta_{7} - \beta_{8} ) q^{46} + ( -24 - 41 \beta_{1} - 22 \beta_{2} + 11 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} - 25 \beta_{7} + 7 \beta_{8} - \beta_{9} ) q^{47} + ( -240 - 25 \beta_{1} - 34 \beta_{2} + 4 \beta_{3} - \beta_{4} + 10 \beta_{5} + \beta_{6} + 17 \beta_{7} - 4 \beta_{8} - \beta_{9} ) q^{48} + ( 3 - 9 \beta_{1} + 13 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 16 \beta_{5} + 2 \beta_{6} + 7 \beta_{7} - 2 \beta_{8} ) q^{49} + ( 22 - 10 \beta_{1} + 18 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 49 \beta_{5} + 4 \beta_{6} + 7 \beta_{7} + 7 \beta_{8} - \beta_{9} ) q^{50} + ( -72 - 72 \beta_{1} - 5 \beta_{2} - 13 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 15 \beta_{7} - 8 \beta_{8} - \beta_{9} ) q^{51} + ( 53 - 40 \beta_{1} + 29 \beta_{2} + 11 \beta_{3} + 3 \beta_{4} - 18 \beta_{5} + 4 \beta_{6} + 5 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{52} + ( -75 - 31 \beta_{1} - 12 \beta_{2} - 3 \beta_{3} + 7 \beta_{4} - \beta_{5} + 7 \beta_{6} + 3 \beta_{7} + 4 \beta_{8} ) q^{53} + ( 31 + 42 \beta_{1} - 15 \beta_{2} + 7 \beta_{3} - \beta_{4} + 34 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} + 6 \beta_{9} ) q^{54} + ( -110 + 17 \beta_{1} - 19 \beta_{2} + 18 \beta_{3} + 5 \beta_{4} + 29 \beta_{5} - 3 \beta_{6} - 20 \beta_{7} + 5 \beta_{8} - 6 \beta_{9} ) q^{55} + ( 84 - 120 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} - 4 \beta_{4} - 25 \beta_{5} + 12 \beta_{6} + 13 \beta_{7} - 7 \beta_{8} + \beta_{9} ) q^{56} + ( 93 - \beta_{1} + 41 \beta_{2} + 13 \beta_{3} + 4 \beta_{4} + 32 \beta_{5} + 6 \beta_{6} - 4 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} ) q^{57} + ( -120 + 13 \beta_{1} - 44 \beta_{2} - 5 \beta_{4} + 40 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} ) q^{58} + ( -174 - 2 \beta_{1} - 55 \beta_{2} - 10 \beta_{3} - 10 \beta_{4} - 35 \beta_{5} + 11 \beta_{6} - 25 \beta_{7} - 8 \beta_{8} + 6 \beta_{9} ) q^{59} + ( 66 + 102 \beta_{1} + 47 \beta_{2} - 4 \beta_{3} + 9 \beta_{4} - 84 \beta_{5} - 5 \beta_{6} + 3 \beta_{7} + 10 \beta_{8} - \beta_{9} ) q^{60} + ( 62 - \beta_{1} + 30 \beta_{2} - \beta_{3} - 10 \beta_{4} + 41 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} - 7 \beta_{8} - 5 \beta_{9} ) q^{61} + ( 125 - 11 \beta_{1} + 28 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 72 \beta_{5} - 7 \beta_{6} - 20 \beta_{7} + 7 \beta_{8} - 12 \beta_{9} ) q^{62} + ( -233 - 93 \beta_{1} - 38 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 45 \beta_{5} - 9 \beta_{6} - 22 \beta_{7} + 8 \beta_{8} - 4 \beta_{9} ) q^{63} + ( -105 + 45 \beta_{1} + 2 \beta_{2} - 11 \beta_{3} - 5 \beta_{4} - 61 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} + 5 \beta_{8} + \beta_{9} ) q^{64} + ( 119 - 45 \beta_{1} - 5 \beta_{2} - 17 \beta_{3} - 12 \beta_{4} + 18 \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} ) q^{65} + ( -185 - 112 \beta_{1} - 32 \beta_{2} + \beta_{3} + 2 \beta_{4} - 76 \beta_{5} - \beta_{6} + 30 \beta_{7} - 10 \beta_{8} - \beta_{9} ) q^{66} + ( 103 - 46 \beta_{1} + 4 \beta_{2} + 5 \beta_{3} + 46 \beta_{5} - 13 \beta_{6} - 3 \beta_{7} + 21 \beta_{8} - 6 \beta_{9} ) q^{67} + ( 105 - 4 \beta_{1} - 17 \beta_{2} - 15 \beta_{3} - 6 \beta_{5} - 3 \beta_{6} - 20 \beta_{7} - 5 \beta_{9} ) q^{68} + ( -333 - 41 \beta_{1} + 13 \beta_{2} + 4 \beta_{3} + 16 \beta_{4} + 25 \beta_{5} - \beta_{6} + 4 \beta_{7} - 4 \beta_{8} - 7 \beta_{9} ) q^{69} + ( 14 + 101 \beta_{1} + 16 \beta_{2} + 10 \beta_{3} + 2 \beta_{4} + 7 \beta_{5} - 12 \beta_{6} - 29 \beta_{7} + 15 \beta_{8} - 5 \beta_{9} ) q^{70} + ( -169 + 118 \beta_{1} - 28 \beta_{2} - 5 \beta_{3} + \beta_{4} + 2 \beta_{5} + 9 \beta_{6} + 13 \beta_{7} + 4 \beta_{8} - 7 \beta_{9} ) q^{71} + ( 375 + 85 \beta_{1} + 17 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 36 \beta_{5} - 21 \beta_{6} - 40 \beta_{7} + 23 \beta_{8} - 6 \beta_{9} ) q^{72} + ( 155 - 59 \beta_{1} + 28 \beta_{2} - 3 \beta_{3} + 12 \beta_{4} + 39 \beta_{5} + 7 \beta_{6} - 11 \beta_{7} - 9 \beta_{8} + 11 \beta_{9} ) q^{73} + ( 219 + 31 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} - 21 \beta_{4} - 16 \beta_{5} + 4 \beta_{6} - 45 \beta_{7} + 14 \beta_{8} - 4 \beta_{9} ) q^{74} + ( 19 - 184 \beta_{1} - 27 \beta_{2} - 3 \beta_{3} - 16 \beta_{4} + 10 \beta_{5} - 16 \beta_{6} - 34 \beta_{7} + 3 \beta_{8} - 6 \beta_{9} ) q^{75} + ( 18 + 143 \beta_{1} - 32 \beta_{2} + 2 \beta_{3} + \beta_{4} + 101 \beta_{5} - 18 \beta_{6} + 5 \beta_{7} + 18 \beta_{8} + 6 \beta_{9} ) q^{76} + ( 112 + 92 \beta_{1} - 53 \beta_{2} + 3 \beta_{3} - 21 \beta_{4} - 30 \beta_{5} + 2 \beta_{6} - 11 \beta_{7} - 4 \beta_{8} + 5 \beta_{9} ) q^{77} + ( 311 - 151 \beta_{1} + 28 \beta_{2} - 11 \beta_{3} - 11 \beta_{4} + 36 \beta_{5} + 20 \beta_{6} - 17 \beta_{7} - 12 \beta_{8} - 2 \beta_{9} ) q^{78} + ( 127 + 58 \beta_{1} - 4 \beta_{2} - 15 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} + 11 \beta_{6} - 15 \beta_{7} - 16 \beta_{8} - 7 \beta_{9} ) q^{79} + ( -266 - 130 \beta_{1} - 43 \beta_{2} - 4 \beta_{3} - 27 \beta_{4} + 92 \beta_{5} + 5 \beta_{6} + \beta_{7} - 4 \beta_{8} - 5 \beta_{9} ) q^{80} + ( -92 + 60 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} + 13 \beta_{4} + 24 \beta_{5} - 18 \beta_{6} - 7 \beta_{7} - 11 \beta_{8} + 7 \beta_{9} ) q^{81} + ( -356 + 47 \beta_{1} - 39 \beta_{2} - 18 \beta_{3} - 7 \beta_{4} - 14 \beta_{5} + 17 \beta_{6} + 13 \beta_{7} - 16 \beta_{8} - \beta_{9} ) q^{82} + ( -4 - 101 \beta_{1} + 25 \beta_{2} + 4 \beta_{3} - 11 \beta_{4} + 58 \beta_{5} - \beta_{6} - 16 \beta_{7} + 21 \beta_{8} ) q^{83} + ( 335 - 103 \beta_{1} + 58 \beta_{2} + 3 \beta_{3} + 8 \beta_{4} - 117 \beta_{5} - 11 \beta_{6} - 19 \beta_{7} + 19 \beta_{8} - 18 \beta_{9} ) q^{84} + ( -152 + 111 \beta_{1} - 13 \beta_{2} + 22 \beta_{3} + 17 \beta_{4} - 59 \beta_{5} + 19 \beta_{6} + 25 \beta_{7} - 15 \beta_{8} + 8 \beta_{9} ) q^{85} + ( 121 + 143 \beta_{1} + 5 \beta_{2} + \beta_{3} - 8 \beta_{4} - 43 \beta_{5} - 26 \beta_{6} - 27 \beta_{7} + 21 \beta_{8} + 5 \beta_{9} ) q^{87} + ( 222 + 108 \beta_{1} - 32 \beta_{2} - 16 \beta_{3} - 16 \beta_{4} - 89 \beta_{5} - 6 \beta_{6} - 23 \beta_{7} + 21 \beta_{8} - \beta_{9} ) q^{88} + ( -191 - 110 \beta_{1} + 29 \beta_{2} + 25 \beta_{3} + 7 \beta_{4} + 20 \beta_{5} - 14 \beta_{6} + 11 \beta_{7} + 12 \beta_{8} - \beta_{9} ) q^{89} + ( -56 - 246 \beta_{1} - 60 \beta_{2} - 32 \beta_{3} - 14 \beta_{4} + 30 \beta_{5} + 14 \beta_{6} + 24 \beta_{7} - 30 \beta_{8} + 8 \beta_{9} ) q^{90} + ( -332 + 48 \beta_{1} - 22 \beta_{2} - 14 \beta_{3} + 2 \beta_{4} + 41 \beta_{5} - 8 \beta_{6} - 2 \beta_{7} + 6 \beta_{8} + 6 \beta_{9} ) q^{91} + ( -277 + 35 \beta_{1} + 8 \beta_{2} - 7 \beta_{3} + 14 \beta_{4} - 35 \beta_{5} - 6 \beta_{6} - 12 \beta_{7} + 4 \beta_{8} ) q^{92} + ( -154 - 259 \beta_{1} - 53 \beta_{2} - 18 \beta_{3} - 23 \beta_{4} - 121 \beta_{5} + 7 \beta_{6} + 23 \beta_{7} - 29 \beta_{8} + 6 \beta_{9} ) q^{93} + ( -440 - 139 \beta_{1} + \beta_{2} + 14 \beta_{3} + 13 \beta_{4} + 79 \beta_{5} + 6 \beta_{6} - \beta_{7} + 4 \beta_{8} + 6 \beta_{9} ) q^{94} + ( -20 - 50 \beta_{1} + 21 \beta_{2} - 16 \beta_{3} + 5 \beta_{4} - 66 \beta_{5} + 5 \beta_{6} + 33 \beta_{7} - 17 \beta_{8} + 11 \beta_{9} ) q^{95} + ( -331 - 320 \beta_{1} - 91 \beta_{2} - 13 \beta_{3} - 19 \beta_{4} + 4 \beta_{5} - 18 \beta_{6} - 15 \beta_{7} - 16 \beta_{8} - 2 \beta_{9} ) q^{96} + ( -49 - 15 \beta_{1} - 11 \beta_{2} - 9 \beta_{3} + 17 \beta_{4} + 18 \beta_{5} + 24 \beta_{6} + 9 \beta_{7} - 32 \beta_{8} - 10 \beta_{9} ) q^{97} + ( -1 + 117 \beta_{1} - 45 \beta_{2} - 25 \beta_{3} - 9 \beta_{4} + 70 \beta_{5} - 10 \beta_{6} + 5 \beta_{7} - 4 \beta_{8} + 10 \beta_{9} ) q^{98} + ( 318 - 176 \beta_{1} - 44 \beta_{2} - 7 \beta_{3} + 3 \beta_{4} - 79 \beta_{5} - 45 \beta_{6} - 42 \beta_{7} + 30 \beta_{8} - 5 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + q^{2} - 5q^{3} + 39q^{4} - 19q^{5} - 15q^{6} - 51q^{7} + 36q^{8} + 117q^{9} + O(q^{10})$$ $$10q + q^{2} - 5q^{3} + 39q^{4} - 19q^{5} - 15q^{6} - 51q^{7} + 36q^{8} + 117q^{9} - 27q^{10} + 27q^{11} - 72q^{12} + 15q^{13} - 96q^{14} - 65q^{15} + 67q^{16} + 82q^{17} + 247q^{18} + 78q^{19} - 495q^{20} - 9q^{21} - 190q^{22} + 61q^{23} - 202q^{24} + 151q^{25} - 21q^{26} + 97q^{27} - 794q^{28} - 53q^{29} + 627q^{30} - 253q^{31} + 399q^{32} - 424q^{33} - 231q^{34} + 355q^{35} + 1092q^{36} - 129q^{37} + 854q^{38} - 691q^{39} - 1345q^{40} + 391q^{41} - 31q^{42} + 377q^{44} - 944q^{45} - 40q^{46} - 334q^{47} - 2401q^{48} + 115q^{49} + 424q^{50} - 795q^{51} + 564q^{52} - 773q^{53} + 182q^{54} - 1242q^{55} + 923q^{56} + 765q^{57} - 1328q^{58} - 1483q^{59} + 1075q^{60} + 437q^{61} + 1509q^{62} - 2222q^{63} - 738q^{64} + 1063q^{65} - 1483q^{66} + 642q^{67} + 1052q^{68} - 3503q^{69} + 85q^{70} - 1545q^{71} + 3834q^{72} + 1292q^{73} + 2232q^{74} - 82q^{75} - 252q^{76} + 1448q^{77} + 2822q^{78} + 1405q^{79} - 3157q^{80} - 974q^{81} - 3304q^{82} - 543q^{83} + 3652q^{84} - 973q^{85} + 1409q^{87} + 2686q^{88} - 2196q^{89} - 742q^{90} - 3513q^{91} - 2629q^{92} - 983q^{93} - 4939q^{94} + 149q^{95} - 3540q^{96} - 425q^{97} - 213q^{98} + 3181q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} - 59 x^{8} + 42 x^{7} + 1187 x^{6} - 541 x^{5} - 9389 x^{4} + 2180 x^{3} + 22676 x^{2} - 320 x - 768$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 12$$ $$\beta_{3}$$ $$=$$ $$($$$$225 \nu^{9} + 1607 \nu^{8} - 30915 \nu^{7} - 48526 \nu^{6} + 948499 \nu^{5} + 145275 \nu^{4} - 8738005 \nu^{3} + 2790076 \nu^{2} + 16689012 \nu - 1568544$$$$)/704512$$ $$\beta_{4}$$ $$=$$ $$($$$$345 \nu^{9} + 8335 \nu^{8} - 47403 \nu^{7} - 385566 \nu^{6} + 1554171 \nu^{5} + 5330467 \nu^{4} - 16950189 \nu^{3} - 23679268 \nu^{2} + 48345556 \nu + 21830112$$$$)/704512$$ $$\beta_{5}$$ $$=$$ $$($$$$-585 \nu^{9} + 225 \nu^{8} + 36347 \nu^{7} - 19138 \nu^{6} - 762059 \nu^{5} + 502925 \nu^{4} + 6140765 \nu^{3} - 3872540 \nu^{2} - 14347924 \nu + 1823776$$$$)/704512$$ $$\beta_{6}$$ $$=$$ $$($$$$-827 \nu^{9} - 3069 \nu^{8} + 51985 \nu^{7} + 139514 \nu^{6} - 1039745 \nu^{5} - 1620089 \nu^{4} + 6893047 \nu^{3} + 2149420 \nu^{2} - 5101372 \nu + 10152032$$$$)/704512$$ $$\beta_{7}$$ $$=$$ $$($$$$-1749 \nu^{9} + 3213 \nu^{8} + 95007 \nu^{7} - 129306 \nu^{6} - 1729711 \nu^{5} + 1424585 \nu^{4} + 12258905 \nu^{3} - 3579884 \nu^{2} - 26340356 \nu + 18848$$$$)/352256$$ $$\beta_{8}$$ $$=$$ $$($$$$-4325 \nu^{9} + 3357 \nu^{8} + 241999 \nu^{7} - 119098 \nu^{6} - 4499167 \nu^{5} + 1229081 \nu^{4} + 32115369 \nu^{3} - 5010348 \nu^{2} - 71167812 \nu + 7371680$$$$)/704512$$ $$\beta_{9}$$ $$=$$ $$($$$$141 \nu^{9} - 213 \nu^{8} - 9191 \nu^{7} + 10586 \nu^{6} + 207911 \nu^{5} - 170401 \nu^{4} - 1838177 \nu^{3} + 949964 \nu^{2} + 4671652 \nu - 704672$$$$)/22016$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 12$$ $$\nu^{3}$$ $$=$$ $$\beta_{8} - \beta_{7} - \beta_{6} + 19 \beta_{1} + 4$$ $$\nu^{4}$$ $$=$$ $$-\beta_{9} + 2 \beta_{8} - 3 \beta_{7} + \beta_{6} - 5 \beta_{5} + \beta_{4} + \beta_{3} + 26 \beta_{2} + 3 \beta_{1} + 229$$ $$\nu^{5}$$ $$=$$ $$\beta_{9} + 33 \beta_{8} - 31 \beta_{7} - 30 \beta_{6} - 10 \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 8 \beta_{2} + 411 \beta_{1} + 165$$ $$\nu^{6}$$ $$=$$ $$-39 \beta_{9} + 85 \beta_{8} - 114 \beta_{7} + 32 \beta_{6} - 261 \beta_{5} + 35 \beta_{4} + 29 \beta_{3} + 658 \beta_{2} + 165 \beta_{1} + 4959$$ $$\nu^{7}$$ $$=$$ $$26 \beta_{9} + 941 \beta_{8} - 858 \beta_{7} - 797 \beta_{6} - 573 \beta_{5} + 89 \beta_{4} - 328 \beta_{3} + 423 \beta_{2} + 9604 \beta_{1} + 5588$$ $$\nu^{8}$$ $$=$$ $$-1214 \beta_{9} + 2809 \beta_{8} - 3510 \beta_{7} + 731 \beta_{6} - 9313 \beta_{5} + 1071 \beta_{4} + 602 \beta_{3} + 16880 \beta_{2} + 6767 \beta_{1} + 116228$$ $$\nu^{9}$$ $$=$$ $$262 \beta_{9} + 25994 \beta_{8} - 23623 \beta_{7} - 20842 \beta_{6} - 23121 \beta_{5} + 3051 \beta_{4} - 11118 \beta_{3} + 16559 \beta_{2} + 236018 \beta_{1} + 177264$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.92559 −3.82341 −3.59365 −1.92278 −0.179767 0.190911 2.14781 3.55840 4.23473 5.31336
−4.92559 −4.99131 16.2615 −2.12330 24.5852 −29.0509 −40.6926 −2.08684 10.4585
1.2 −3.82341 7.76517 6.61843 −18.1323 −29.6894 −5.19796 5.28231 33.2979 69.3271
1.3 −3.59365 1.67792 4.91431 9.72181 −6.02986 15.7603 11.0889 −24.1846 −34.9368
1.4 −1.92278 −8.37832 −4.30290 −0.0702257 16.1097 23.4425 23.6558 43.1963 0.135029
1.5 −0.179767 6.02248 −7.96768 10.9703 −1.08264 −8.78367 2.87047 9.27026 −1.97211
1.6 0.190911 −1.43836 −7.96355 −16.3462 −0.274600 −6.23994 −3.04762 −24.9311 −3.12068
1.7 2.14781 −6.84883 −3.38692 17.1363 −14.7100 −22.9786 −24.4569 19.9064 36.8056
1.8 3.55840 0.389088 4.66219 1.72780 1.38453 21.1861 −11.8773 −26.8486 6.14819
1.9 4.23473 8.85858 9.93292 −5.96488 37.5137 −22.5055 8.18537 51.4744 −25.2597
1.10 5.31336 −8.05642 20.2317 −15.9193 −42.8066 −16.6323 64.9916 37.9059 −84.5852
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$43$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1849.4.a.f 10
43.b odd 2 1 1849.4.a.d 10
43.d odd 6 2 43.4.c.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.4.c.a 20 43.d odd 6 2
1849.4.a.d 10 43.b odd 2 1
1849.4.a.f 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1849))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 21 T^{2} - 30 T^{3} + 291 T^{4} - 493 T^{5} + 3299 T^{6} - 6020 T^{7} + 30260 T^{8} - 57696 T^{9} + 252608 T^{10} - 461568 T^{11} + 1936640 T^{12} - 3082240 T^{13} + 13512704 T^{14} - 16154624 T^{15} + 76283904 T^{16} - 62914560 T^{17} + 352321536 T^{18} - 134217728 T^{19} + 1073741824 T^{20}$$
$3$ $$1 + 5 T + 89 T^{2} + 326 T^{3} + 4555 T^{4} + 14043 T^{5} + 192137 T^{6} + 556492 T^{7} + 6664441 T^{8} + 16945357 T^{9} + 192259862 T^{10} + 457524639 T^{11} + 4858377489 T^{12} + 10953432036 T^{13} + 102109479417 T^{14} + 201501701001 T^{15} + 1764700327395 T^{16} + 3410075144178 T^{17} + 25136228746809 T^{18} + 38127987424935 T^{19} + 205891132094649 T^{20}$$
$5$ $$1 + 19 T + 730 T^{2} + 11645 T^{3} + 265090 T^{4} + 3590441 T^{5} + 63812448 T^{6} + 751352097 T^{7} + 11330307869 T^{8} + 118903300014 T^{9} + 1578548875564 T^{10} + 14862912501750 T^{11} + 177036060453125 T^{12} + 1467484564453125 T^{13} + 15579210937500000 T^{14} + 109571563720703125 T^{15} + 1011238098144531250 T^{16} + 5552768707275390625 T^{17} + 43511390686035156250 T^{18} +$$$$14\!\cdots\!75$$$$T^{19} +$$$$93\!\cdots\!25$$$$T^{20}$$
$7$ $$1 + 51 T + 2958 T^{2} + 98283 T^{3} + 3538173 T^{4} + 93650638 T^{5} + 2648528462 T^{6} + 59561803592 T^{7} + 1408530194217 T^{8} + 27332735053531 T^{9} + 555667189834924 T^{10} + 9375128123361133 T^{11} + 165712168819435833 T^{12} + 2403533614362756344 T^{13} + 36659043102564814862 T^{14} +$$$$44\!\cdots\!34$$$$T^{15} +$$$$57\!\cdots\!77$$$$T^{16} +$$$$54\!\cdots\!81$$$$T^{17} +$$$$56\!\cdots\!58$$$$T^{18} +$$$$33\!\cdots\!93$$$$T^{19} +$$$$22\!\cdots\!49$$$$T^{20}$$
$11$ $$1 - 27 T + 5077 T^{2} - 61984 T^{3} + 15043654 T^{4} - 162363748 T^{5} + 35128969454 T^{6} - 264446154846 T^{7} + 61397630693497 T^{8} - 405185649214743 T^{9} + 91897922249805306 T^{10} - 539302099104822933 T^{11} +$$$$10\!\cdots\!17$$$$T^{12} -$$$$62\!\cdots\!86$$$$T^{13} +$$$$11\!\cdots\!34$$$$T^{14} -$$$$67\!\cdots\!48$$$$T^{15} +$$$$83\!\cdots\!74$$$$T^{16} -$$$$45\!\cdots\!24$$$$T^{17} +$$$$50\!\cdots\!57$$$$T^{18} -$$$$35\!\cdots\!17$$$$T^{19} +$$$$17\!\cdots\!01$$$$T^{20}$$
$13$ $$1 - 15 T + 16061 T^{2} - 162762 T^{3} + 119085587 T^{4} - 746573887 T^{5} + 549572425527 T^{6} - 1904744585888 T^{7} + 1797963317266135 T^{8} - 3466193660344645 T^{9} + 4474548970144863094 T^{10} - 7615227471777185065 T^{11} +$$$$86\!\cdots\!15$$$$T^{12} -$$$$20\!\cdots\!24$$$$T^{13} +$$$$12\!\cdots\!87$$$$T^{14} -$$$$38\!\cdots\!59$$$$T^{15} +$$$$13\!\cdots\!23$$$$T^{16} -$$$$40\!\cdots\!06$$$$T^{17} +$$$$87\!\cdots\!21$$$$T^{18} -$$$$17\!\cdots\!55$$$$T^{19} +$$$$26\!\cdots\!49$$$$T^{20}$$
$17$ $$1 - 82 T + 31542 T^{2} - 2729726 T^{3} + 495433616 T^{4} - 43592811872 T^{5} + 5079597607931 T^{6} - 435855391738836 T^{7} + 37675035599844963 T^{8} - 3000826927588845966 T^{9} +$$$$21\!\cdots\!17$$$$T^{10} -$$$$14\!\cdots\!58$$$$T^{11} +$$$$90\!\cdots\!47$$$$T^{12} -$$$$51\!\cdots\!92$$$$T^{13} +$$$$29\!\cdots\!91$$$$T^{14} -$$$$12\!\cdots\!96$$$$T^{15} +$$$$69\!\cdots\!44$$$$T^{16} -$$$$18\!\cdots\!42$$$$T^{17} +$$$$10\!\cdots\!82$$$$T^{18} -$$$$13\!\cdots\!86$$$$T^{19} +$$$$81\!\cdots\!49$$$$T^{20}$$
$19$ $$1 - 78 T + 36391 T^{2} - 2240906 T^{3} + 592121360 T^{4} - 29939098218 T^{5} + 6238019086681 T^{6} - 285492506123062 T^{7} + 52918040345784791 T^{8} - 2341393657746342360 T^{9} +$$$$39\!\cdots\!04$$$$T^{10} -$$$$16\!\cdots\!40$$$$T^{11} +$$$$24\!\cdots\!71$$$$T^{12} -$$$$92\!\cdots\!98$$$$T^{13} +$$$$13\!\cdots\!41$$$$T^{14} -$$$$45\!\cdots\!82$$$$T^{15} +$$$$61\!\cdots\!60$$$$T^{16} -$$$$16\!\cdots\!14$$$$T^{17} +$$$$17\!\cdots\!11$$$$T^{18} -$$$$26\!\cdots\!42$$$$T^{19} +$$$$23\!\cdots\!01$$$$T^{20}$$
$23$ $$1 - 61 T + 85545 T^{2} - 5735688 T^{3} + 3578624869 T^{4} - 244193475031 T^{5} + 96258657044673 T^{6} - 6286223931296694 T^{7} + 1836703681458483033 T^{8} -$$$$10\!\cdots\!83$$$$T^{9} +$$$$25\!\cdots\!30$$$$T^{10} -$$$$13\!\cdots\!61$$$$T^{11} +$$$$27\!\cdots\!37$$$$T^{12} -$$$$11\!\cdots\!22$$$$T^{13} +$$$$21\!\cdots\!33$$$$T^{14} -$$$$65\!\cdots\!17$$$$T^{15} +$$$$11\!\cdots\!61$$$$T^{16} -$$$$22\!\cdots\!24$$$$T^{17} +$$$$41\!\cdots\!45$$$$T^{18} -$$$$35\!\cdots\!67$$$$T^{19} +$$$$71\!\cdots\!49$$$$T^{20}$$
$29$ $$1 + 53 T + 141138 T^{2} + 8023609 T^{3} + 9724245343 T^{4} + 610047550838 T^{5} + 443117344967692 T^{6} + 30265015746923862 T^{7} + 15138818067933540037 T^{8} +$$$$10\!\cdots\!31$$$$T^{9} +$$$$41\!\cdots\!30$$$$T^{10} +$$$$25\!\cdots\!59$$$$T^{11} +$$$$90\!\cdots\!77$$$$T^{12} +$$$$43\!\cdots\!78$$$$T^{13} +$$$$15\!\cdots\!72$$$$T^{14} +$$$$52\!\cdots\!62$$$$T^{15} +$$$$20\!\cdots\!23$$$$T^{16} +$$$$41\!\cdots\!61$$$$T^{17} +$$$$17\!\cdots\!78$$$$T^{18} +$$$$16\!\cdots\!77$$$$T^{19} +$$$$74\!\cdots\!01$$$$T^{20}$$
$31$ $$1 + 253 T + 105884 T^{2} + 17011687 T^{3} + 6345862600 T^{4} + 759122832695 T^{5} + 238432130617916 T^{6} + 21648160854623909 T^{7} + 8562912517186200751 T^{8} +$$$$64\!\cdots\!12$$$$T^{9} +$$$$24\!\cdots\!36$$$$T^{10} +$$$$19\!\cdots\!92$$$$T^{11} +$$$$75\!\cdots\!31$$$$T^{12} +$$$$57\!\cdots\!39$$$$T^{13} +$$$$18\!\cdots\!76$$$$T^{14} +$$$$17\!\cdots\!45$$$$T^{15} +$$$$44\!\cdots\!00$$$$T^{16} +$$$$35\!\cdots\!97$$$$T^{17} +$$$$65\!\cdots\!64$$$$T^{18} +$$$$46\!\cdots\!83$$$$T^{19} +$$$$55\!\cdots\!01$$$$T^{20}$$
$37$ $$1 + 129 T + 285619 T^{2} + 40107546 T^{3} + 41430046973 T^{4} + 5853675673369 T^{5} + 4034937015391223 T^{6} + 546994538186490476 T^{7} +$$$$29\!\cdots\!87$$$$T^{8} +$$$$36\!\cdots\!35$$$$T^{9} +$$$$16\!\cdots\!36$$$$T^{10} +$$$$18\!\cdots\!55$$$$T^{11} +$$$$75\!\cdots\!83$$$$T^{12} +$$$$71\!\cdots\!52$$$$T^{13} +$$$$26\!\cdots\!63$$$$T^{14} +$$$$19\!\cdots\!17$$$$T^{15} +$$$$69\!\cdots\!17$$$$T^{16} +$$$$34\!\cdots\!02$$$$T^{17} +$$$$12\!\cdots\!59$$$$T^{18} +$$$$28\!\cdots\!57$$$$T^{19} +$$$$11\!\cdots\!49$$$$T^{20}$$
$41$ $$1 - 391 T + 466858 T^{2} - 159315937 T^{3} + 107799629054 T^{4} - 32107569594125 T^{5} + 16028988094146372 T^{6} - 4181844238111783617 T^{7} +$$$$16\!\cdots\!81$$$$T^{8} -$$$$38\!\cdots\!70$$$$T^{9} +$$$$13\!\cdots\!96$$$$T^{10} -$$$$26\!\cdots\!70$$$$T^{11} +$$$$80\!\cdots\!21$$$$T^{12} -$$$$13\!\cdots\!37$$$$T^{13} +$$$$36\!\cdots\!32$$$$T^{14} -$$$$49\!\cdots\!25$$$$T^{15} +$$$$11\!\cdots\!34$$$$T^{16} -$$$$11\!\cdots\!17$$$$T^{17} +$$$$23\!\cdots\!38$$$$T^{18} -$$$$13\!\cdots\!71$$$$T^{19} +$$$$24\!\cdots\!01$$$$T^{20}$$
$43$ 1
$47$ $$1 + 334 T + 469033 T^{2} + 125686554 T^{3} + 97290890628 T^{4} + 20127660597266 T^{5} + 12996286592743642 T^{6} + 2230581953572455798 T^{7} +$$$$14\!\cdots\!19$$$$T^{8} +$$$$24\!\cdots\!68$$$$T^{9} +$$$$16\!\cdots\!34$$$$T^{10} +$$$$25\!\cdots\!64$$$$T^{11} +$$$$15\!\cdots\!51$$$$T^{12} +$$$$24\!\cdots\!66$$$$T^{13} +$$$$15\!\cdots\!22$$$$T^{14} +$$$$24\!\cdots\!38$$$$T^{15} +$$$$12\!\cdots\!92$$$$T^{16} +$$$$16\!\cdots\!38$$$$T^{17} +$$$$63\!\cdots\!73$$$$T^{18} +$$$$46\!\cdots\!42$$$$T^{19} +$$$$14\!\cdots\!49$$$$T^{20}$$
$53$ $$1 + 773 T + 1139233 T^{2} + 720257454 T^{3} + 606121338023 T^{4} + 324763897859933 T^{5} + 202760779719698827 T^{6} + 93491859975188097460 T^{7} +$$$$47\!\cdots\!43$$$$T^{8} +$$$$18\!\cdots\!79$$$$T^{9} +$$$$82\!\cdots\!10$$$$T^{10} +$$$$28\!\cdots\!83$$$$T^{11} +$$$$10\!\cdots\!47$$$$T^{12} +$$$$30\!\cdots\!80$$$$T^{13} +$$$$99\!\cdots\!07$$$$T^{14} +$$$$23\!\cdots\!81$$$$T^{15} +$$$$65\!\cdots\!47$$$$T^{16} +$$$$11\!\cdots\!62$$$$T^{17} +$$$$27\!\cdots\!73$$$$T^{18} +$$$$27\!\cdots\!01$$$$T^{19} +$$$$53\!\cdots\!49$$$$T^{20}$$
$59$ $$1 + 1483 T + 1378157 T^{2} + 969130766 T^{3} + 686264212828 T^{4} + 451249739385536 T^{5} + 271319879713111662 T^{6} +$$$$14\!\cdots\!74$$$$T^{7} +$$$$73\!\cdots\!15$$$$T^{8} +$$$$36\!\cdots\!33$$$$T^{9} +$$$$17\!\cdots\!86$$$$T^{10} +$$$$74\!\cdots\!07$$$$T^{11} +$$$$30\!\cdots\!15$$$$T^{12} +$$$$12\!\cdots\!86$$$$T^{13} +$$$$48\!\cdots\!22$$$$T^{14} +$$$$16\!\cdots\!64$$$$T^{15} +$$$$51\!\cdots\!88$$$$T^{16} +$$$$14\!\cdots\!94$$$$T^{17} +$$$$43\!\cdots\!77$$$$T^{18} +$$$$96\!\cdots\!77$$$$T^{19} +$$$$13\!\cdots\!01$$$$T^{20}$$
$61$ $$1 - 437 T + 935885 T^{2} - 416954406 T^{3} + 554190861587 T^{4} - 216661010493785 T^{5} + 229499018020120339 T^{6} - 83061468261243730036 T^{7} +$$$$72\!\cdots\!59$$$$T^{8} -$$$$23\!\cdots\!99$$$$T^{9} +$$$$18\!\cdots\!58$$$$T^{10} -$$$$53\!\cdots\!19$$$$T^{11} +$$$$37\!\cdots\!99$$$$T^{12} -$$$$97\!\cdots\!76$$$$T^{13} +$$$$60\!\cdots\!19$$$$T^{14} -$$$$13\!\cdots\!85$$$$T^{15} +$$$$75\!\cdots\!47$$$$T^{16} -$$$$12\!\cdots\!66$$$$T^{17} +$$$$65\!\cdots\!85$$$$T^{18} -$$$$69\!\cdots\!77$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20}$$
$67$ $$1 - 642 T + 2067386 T^{2} - 1339263368 T^{3} + 2018557242358 T^{4} - 1330913370861868 T^{5} + 1251783460266323431 T^{6} -$$$$82\!\cdots\!10$$$$T^{7} +$$$$55\!\cdots\!99$$$$T^{8} -$$$$34\!\cdots\!00$$$$T^{9} +$$$$18\!\cdots\!89$$$$T^{10} -$$$$10\!\cdots\!00$$$$T^{11} +$$$$50\!\cdots\!31$$$$T^{12} -$$$$22\!\cdots\!70$$$$T^{13} +$$$$10\!\cdots\!91$$$$T^{14} -$$$$32\!\cdots\!24$$$$T^{15} +$$$$14\!\cdots\!22$$$$T^{16} -$$$$29\!\cdots\!56$$$$T^{17} +$$$$13\!\cdots\!06$$$$T^{18} -$$$$12\!\cdots\!66$$$$T^{19} +$$$$60\!\cdots\!49$$$$T^{20}$$
$71$ $$1 + 1545 T + 2551637 T^{2} + 2382888830 T^{3} + 2318471332835 T^{4} + 1505208541247391 T^{5} + 1076364279273182533 T^{6} +$$$$50\!\cdots\!52$$$$T^{7} +$$$$31\!\cdots\!65$$$$T^{8} +$$$$11\!\cdots\!05$$$$T^{9} +$$$$90\!\cdots\!10$$$$T^{10} +$$$$42\!\cdots\!55$$$$T^{11} +$$$$40\!\cdots\!65$$$$T^{12} +$$$$23\!\cdots\!12$$$$T^{13} +$$$$17\!\cdots\!53$$$$T^{14} +$$$$88\!\cdots\!41$$$$T^{15} +$$$$48\!\cdots\!35$$$$T^{16} +$$$$17\!\cdots\!30$$$$T^{17} +$$$$68\!\cdots\!97$$$$T^{18} +$$$$14\!\cdots\!95$$$$T^{19} +$$$$34\!\cdots\!01$$$$T^{20}$$
$73$ $$1 - 1292 T + 2470024 T^{2} - 2540022046 T^{3} + 3113964531230 T^{4} - 2703184247613182 T^{5} + 2581291938334464755 T^{6} -$$$$19\!\cdots\!84$$$$T^{7} +$$$$15\!\cdots\!31$$$$T^{8} -$$$$10\!\cdots\!52$$$$T^{9} +$$$$68\!\cdots\!47$$$$T^{10} -$$$$38\!\cdots\!84$$$$T^{11} +$$$$23\!\cdots\!59$$$$T^{12} -$$$$11\!\cdots\!92$$$$T^{13} +$$$$59\!\cdots\!55$$$$T^{14} -$$$$24\!\cdots\!74$$$$T^{15} +$$$$10\!\cdots\!70$$$$T^{16} -$$$$34\!\cdots\!58$$$$T^{17} +$$$$12\!\cdots\!84$$$$T^{18} -$$$$26\!\cdots\!24$$$$T^{19} +$$$$79\!\cdots\!49$$$$T^{20}$$
$79$ $$1 - 1405 T + 4124293 T^{2} - 4455739938 T^{3} + 7389320412515 T^{4} - 6538402276888679 T^{5} + 7995580193148085185 T^{6} -$$$$60\!\cdots\!84$$$$T^{7} +$$$$60\!\cdots\!69$$$$T^{8} -$$$$39\!\cdots\!21$$$$T^{9} +$$$$34\!\cdots\!50$$$$T^{10} -$$$$19\!\cdots\!19$$$$T^{11} +$$$$14\!\cdots\!49$$$$T^{12} -$$$$72\!\cdots\!96$$$$T^{13} +$$$$47\!\cdots\!85$$$$T^{14} -$$$$19\!\cdots\!21$$$$T^{15} +$$$$10\!\cdots\!15$$$$T^{16} -$$$$31\!\cdots\!02$$$$T^{17} +$$$$14\!\cdots\!33$$$$T^{18} -$$$$24\!\cdots\!95$$$$T^{19} +$$$$84\!\cdots\!01$$$$T^{20}$$
$83$ $$1 + 543 T + 3505821 T^{2} + 2026676938 T^{3} + 6084107600779 T^{4} + 3508580210105525 T^{5} + 6967823140297410865 T^{6} +$$$$38\!\cdots\!20$$$$T^{7} +$$$$58\!\cdots\!89$$$$T^{8} +$$$$29\!\cdots\!71$$$$T^{9} +$$$$37\!\cdots\!50$$$$T^{10} +$$$$17\!\cdots\!77$$$$T^{11} +$$$$19\!\cdots\!41$$$$T^{12} +$$$$71\!\cdots\!60$$$$T^{13} +$$$$74\!\cdots\!65$$$$T^{14} +$$$$21\!\cdots\!75$$$$T^{15} +$$$$21\!\cdots\!11$$$$T^{16} +$$$$40\!\cdots\!54$$$$T^{17} +$$$$40\!\cdots\!41$$$$T^{18} +$$$$35\!\cdots\!61$$$$T^{19} +$$$$37\!\cdots\!49$$$$T^{20}$$
$89$ $$1 + 2196 T + 7101672 T^{2} + 11169690002 T^{3} + 21314174237118 T^{4} + 26581378923914546 T^{5} + 37764446306913683967 T^{6} +$$$$39\!\cdots\!68$$$$T^{7} +$$$$44\!\cdots\!03$$$$T^{8} +$$$$38\!\cdots\!96$$$$T^{9} +$$$$37\!\cdots\!75$$$$T^{10} +$$$$27\!\cdots\!24$$$$T^{11} +$$$$22\!\cdots\!83$$$$T^{12} +$$$$13\!\cdots\!12$$$$T^{13} +$$$$93\!\cdots\!07$$$$T^{14} +$$$$46\!\cdots\!54$$$$T^{15} +$$$$26\!\cdots\!58$$$$T^{16} +$$$$96\!\cdots\!78$$$$T^{17} +$$$$43\!\cdots\!52$$$$T^{18} +$$$$94\!\cdots\!84$$$$T^{19} +$$$$30\!\cdots\!01$$$$T^{20}$$
$97$ $$1 + 425 T + 4855100 T^{2} + 2674736785 T^{3} + 12413935569789 T^{4} + 7844309405033348 T^{5} + 21709474743257501488 T^{6} +$$$$14\!\cdots\!84$$$$T^{7} +$$$$28\!\cdots\!94$$$$T^{8} +$$$$18\!\cdots\!26$$$$T^{9} +$$$$29\!\cdots\!88$$$$T^{10} +$$$$16\!\cdots\!98$$$$T^{11} +$$$$23\!\cdots\!26$$$$T^{12} +$$$$10\!\cdots\!28$$$$T^{13} +$$$$15\!\cdots\!08$$$$T^{14} +$$$$49\!\cdots\!64$$$$T^{15} +$$$$71\!\cdots\!21$$$$T^{16} +$$$$14\!\cdots\!45$$$$T^{17} +$$$$23\!\cdots\!00$$$$T^{18} +$$$$18\!\cdots\!25$$$$T^{19} +$$$$40\!\cdots\!49$$$$T^{20}$$