# Properties

 Label 16.4.e.a Level 16 Weight 4 Character orbit 16.e Analytic conductor 0.944 Analytic rank 0 Dimension 10 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 16.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.944030560092$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} -\beta_{5} q^{3} + ( 1 - \beta_{2} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{4} + ( -1 + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{5} + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{6} + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{7} + ( -6 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{8} + ( 1 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{2} -\beta_{5} q^{3} + ( 1 - \beta_{2} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{4} + ( -1 + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{5} + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{6} + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{7} + ( -6 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{8} + ( 1 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{9} + ( -6 - 8 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{10} + ( 2 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - 6 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{11} + ( 10 + 2 \beta_{1} + 16 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{12} + ( 2 - \beta_{1} + \beta_{2} - 9 \beta_{4} + 4 \beta_{5} + 7 \beta_{6} - \beta_{7} + \beta_{8} + 4 \beta_{9} ) q^{13} + ( 20 + 2 \beta_{1} - 14 \beta_{2} - 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 4 \beta_{9} ) q^{14} + ( -9 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} - 14 \beta_{4} - \beta_{5} + 4 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} ) q^{15} + ( 28 - 6 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{16} + ( -3 + 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 16 \beta_{6} - 2 \beta_{8} - 5 \beta_{9} ) q^{17} + ( 14 - 2 \beta_{1} + 50 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 6 \beta_{9} ) q^{18} + ( 8 \beta_{1} + 10 \beta_{2} + 10 \beta_{4} + 3 \beta_{5} + 18 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{19} + ( -22 + 8 \beta_{1} - 48 \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 9 \beta_{6} - \beta_{7} - 5 \beta_{8} + 4 \beta_{9} ) q^{20} + ( 2 - 4 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} + 20 \beta_{4} - 4 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} ) q^{21} + ( -59 - \beta_{1} + 11 \beta_{2} + \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 7 \beta_{7} + 5 \beta_{8} - 5 \beta_{9} ) q^{22} + ( -7 - 7 \beta_{1} - 27 \beta_{2} + 3 \beta_{3} + 12 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} - 10 \beta_{7} + \beta_{9} ) q^{23} + ( -80 - 2 \beta_{1} + 28 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 12 \beta_{6} + 6 \beta_{7} - 4 \beta_{8} - 6 \beta_{9} ) q^{24} + ( 5 \beta_{2} - 12 \beta_{3} + 16 \beta_{4} - 12 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} - 8 \beta_{9} ) q^{25} + ( -28 - 6 \beta_{1} - 74 \beta_{2} - 10 \beta_{3} + 5 \beta_{4} + 8 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} + 4 \beta_{9} ) q^{26} + ( 14 + 6 \beta_{1} + 16 \beta_{2} + 4 \beta_{3} - 6 \beta_{4} - 18 \beta_{6} - 10 \beta_{7} - 10 \beta_{8} + 8 \beta_{9} ) q^{27} + ( 34 + 8 \beta_{1} + 94 \beta_{2} + 10 \beta_{3} - 22 \beta_{4} + 4 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} ) q^{28} + ( -16 - 3 \beta_{1} + 13 \beta_{2} - 11 \beta_{4} - 24 \beta_{5} - 11 \beta_{6} + 5 \beta_{7} - 5 \beta_{8} ) q^{29} + ( 122 - 40 \beta_{2} - 6 \beta_{3} + 10 \beta_{4} + 20 \beta_{5} + 14 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} - 2 \beta_{9} ) q^{30} + ( 34 + 2 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} - 12 \beta_{8} + 6 \beta_{9} ) q^{31} + ( 96 + 4 \beta_{1} - 64 \beta_{2} - 6 \beta_{3} - 30 \beta_{4} + 6 \beta_{5} - 14 \beta_{6} - 2 \beta_{7} - 10 \beta_{8} ) q^{32} + ( 7 - 3 \beta_{1} - 3 \beta_{2} + 14 \beta_{3} - 2 \beta_{4} - 14 \beta_{5} + 10 \beta_{8} - 3 \beta_{9} ) q^{33} + ( 50 - 2 \beta_{1} + 106 \beta_{2} + 18 \beta_{3} - 12 \beta_{4} - 2 \beta_{5} + 6 \beta_{6} - 14 \beta_{7} + 10 \beta_{8} + 2 \beta_{9} ) q^{34} + ( 48 - 8 \beta_{1} - 52 \beta_{2} - 12 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} - 12 \beta_{7} + 12 \beta_{8} + 4 \beta_{9} ) q^{35} + ( -43 - 4 \beta_{1} - 131 \beta_{2} + 3 \beta_{3} - 19 \beta_{4} - 20 \beta_{5} + 36 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} ) q^{36} + ( -5 + 4 \beta_{1} - 10 \beta_{2} - 28 \beta_{3} - 9 \beta_{4} - 7 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{37} + ( -127 + \beta_{1} + 97 \beta_{2} - 23 \beta_{3} + 24 \beta_{4} - 19 \beta_{5} - 4 \beta_{6} + 17 \beta_{7} - 5 \beta_{8} + 3 \beta_{9} ) q^{38} + ( 11 + 11 \beta_{1} + 79 \beta_{2} + 7 \beta_{3} - 12 \beta_{4} + 7 \beta_{5} + 22 \beta_{6} + 10 \beta_{7} - 5 \beta_{9} ) q^{39} + ( -144 + 2 \beta_{1} + 52 \beta_{2} + 12 \beta_{3} + 16 \beta_{4} - 14 \beta_{5} - 54 \beta_{6} - 8 \beta_{7} - 2 \beta_{8} + 10 \beta_{9} ) q^{40} + ( 2 + 2 \beta_{1} + 6 \beta_{2} + 24 \beta_{3} - 16 \beta_{4} + 24 \beta_{5} - 12 \beta_{6} + 4 \beta_{7} + 6 \beta_{9} ) q^{41} + ( -76 + 4 \beta_{1} - 156 \beta_{2} - 12 \beta_{3} - 6 \beta_{4} - 12 \beta_{5} - 18 \beta_{6} - 12 \beta_{7} - 20 \beta_{8} - 4 \beta_{9} ) q^{42} + ( -72 - 8 \beta_{1} - 80 \beta_{2} + 3 \beta_{3} - 8 \beta_{4} + 40 \beta_{6} + 8 \beta_{7} + 8 \beta_{8} - 16 \beta_{9} ) q^{43} + ( 72 - 14 \beta_{1} + 134 \beta_{2} + 5 \beta_{3} + 63 \beta_{4} + 19 \beta_{5} + 27 \beta_{6} - 9 \beta_{7} - \beta_{8} + 6 \beta_{9} ) q^{44} + ( 2 + 5 \beta_{1} - 9 \beta_{2} + 37 \beta_{4} + 44 \beta_{5} - 11 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 12 \beta_{9} ) q^{45} + ( 104 - 10 \beta_{1} - 34 \beta_{2} + 20 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - 30 \beta_{6} - 12 \beta_{7} - 22 \beta_{8} + 16 \beta_{9} ) q^{46} + ( -102 + 10 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 36 \beta_{4} + 4 \beta_{5} - 24 \beta_{6} + 20 \beta_{8} - 2 \beta_{9} ) q^{47} + ( 168 + 16 \beta_{1} - 16 \beta_{2} - 26 \beta_{3} + 86 \beta_{4} - 2 \beta_{5} + 34 \beta_{6} + 10 \beta_{7} + 6 \beta_{8} + 4 \beta_{9} ) q^{48} + ( 1 - 10 \beta_{1} + 22 \beta_{2} - 36 \beta_{3} + 20 \beta_{4} + 36 \beta_{5} + 64 \beta_{6} - 4 \beta_{8} + 22 \beta_{9} ) q^{49} + ( 80 + 8 \beta_{1} + 120 \beta_{2} + 8 \beta_{4} - 24 \beta_{5} - 11 \beta_{6} + 32 \beta_{7} - 8 \beta_{8} - 32 \beta_{9} ) q^{50} + ( -160 - 16 \beta_{1} + 130 \beta_{2} - 22 \beta_{4} - 22 \beta_{5} - 78 \beta_{6} + 18 \beta_{7} - 18 \beta_{8} - 14 \beta_{9} ) q^{51} + ( -56 - 28 \beta_{1} - 122 \beta_{2} - 21 \beta_{3} + 41 \beta_{4} - \beta_{5} - 85 \beta_{6} - 3 \beta_{7} + 15 \beta_{8} - 16 \beta_{9} ) q^{52} + ( -5 + 12 \beta_{1} - 38 \beta_{2} + 68 \beta_{3} - 85 \beta_{4} + 37 \beta_{6} - 19 \beta_{7} - 19 \beta_{8} - 21 \beta_{9} ) q^{53} + ( -120 + 4 \beta_{1} + 12 \beta_{2} + 44 \beta_{3} - 36 \beta_{4} + 24 \beta_{6} - 12 \beta_{7} - 8 \beta_{8} + 24 \beta_{9} ) q^{54} + ( 15 + 15 \beta_{1} - 157 \beta_{2} - 25 \beta_{3} - 60 \beta_{4} - 25 \beta_{5} - 18 \beta_{6} + 18 \beta_{7} + 15 \beta_{9} ) q^{55} + ( -20 + 16 \beta_{1} + 60 \beta_{2} - 14 \beta_{3} - 38 \beta_{4} + 14 \beta_{5} + 114 \beta_{6} - 18 \beta_{7} + 22 \beta_{8} + 12 \beta_{9} ) q^{56} + ( -9 - 9 \beta_{1} - 21 \beta_{2} - 18 \beta_{3} - 64 \beta_{4} - 18 \beta_{5} - 54 \beta_{6} - 46 \beta_{7} + 41 \beta_{9} ) q^{57} + ( 16 + 34 \beta_{1} - 106 \beta_{2} + 30 \beta_{3} - 3 \beta_{4} + 20 \beta_{5} + 19 \beta_{6} + 18 \beta_{7} - 8 \beta_{8} - 24 \beta_{9} ) q^{58} + ( 164 - 36 \beta_{1} + 192 \beta_{2} - 29 \beta_{3} + 84 \beta_{4} + 60 \beta_{6} + 28 \beta_{7} + 28 \beta_{8} - 8 \beta_{9} ) q^{59} + ( -22 - 36 \beta_{1} + 58 \beta_{2} - 44 \beta_{3} - 96 \beta_{4} - 14 \beta_{5} - 50 \beta_{6} - 12 \beta_{7} + 18 \beta_{8} ) q^{60} + ( 94 + 19 \beta_{1} - 79 \beta_{2} + 67 \beta_{4} - 12 \beta_{5} + 51 \beta_{6} - 21 \beta_{7} + 21 \beta_{8} - 4 \beta_{9} ) q^{61} + ( 12 + 4 \beta_{1} - 68 \beta_{2} + 4 \beta_{3} - 36 \beta_{4} - 36 \beta_{5} - 4 \beta_{6} + 20 \beta_{7} + 4 \beta_{8} + 12 \beta_{9} ) q^{62} + ( 267 + 15 \beta_{1} - 7 \beta_{2} + 33 \beta_{3} + 26 \beta_{4} - 33 \beta_{5} - 44 \beta_{6} + 10 \beta_{8} - 7 \beta_{9} ) q^{63} + ( 80 - 4 \beta_{1} + 32 \beta_{2} + 28 \beta_{3} - 116 \beta_{4} - 32 \beta_{6} - 4 \beta_{7} + 40 \beta_{8} + 12 \beta_{9} ) q^{64} + ( -44 + 18 \beta_{1} + 26 \beta_{2} + 40 \beta_{3} + 36 \beta_{4} - 40 \beta_{5} + 16 \beta_{6} - 52 \beta_{8} + 26 \beta_{9} ) q^{65} + ( -58 + 18 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} + 50 \beta_{5} + 2 \beta_{6} - 10 \beta_{7} - 26 \beta_{8} - 10 \beta_{9} ) q^{66} + ( 176 - 202 \beta_{2} + 30 \beta_{4} - 13 \beta_{5} - 74 \beta_{6} + 22 \beta_{7} - 22 \beta_{8} - 26 \beta_{9} ) q^{67} + ( -78 - 4 \beta_{1} + 38 \beta_{2} - 4 \beta_{3} - 36 \beta_{4} + 66 \beta_{5} + 130 \beta_{6} - 20 \beta_{7} - 22 \beta_{8} + 20 \beta_{9} ) q^{68} + ( 58 - 12 \beta_{1} + 54 \beta_{2} - 52 \beta_{3} + 48 \beta_{6} + 8 \beta_{7} + 8 \beta_{8} - 16 \beta_{9} ) q^{69} + ( 2 - 30 \beta_{1} - 62 \beta_{2} + 2 \beta_{3} - 40 \beta_{4} + 42 \beta_{5} - 32 \beta_{6} - 14 \beta_{7} + 6 \beta_{8} + 6 \beta_{9} ) q^{70} + ( 1 + \beta_{1} + 333 \beta_{2} - 21 \beta_{3} - 20 \beta_{4} - 21 \beta_{5} + 10 \beta_{6} - 26 \beta_{7} + 9 \beta_{9} ) q^{71} + ( 70 + 24 \beta_{1} - 14 \beta_{2} - 33 \beta_{3} + 79 \beta_{4} + 45 \beta_{5} - 113 \beta_{6} + 49 \beta_{7} + \beta_{8} - 26 \beta_{9} ) q^{72} + ( -29 - 29 \beta_{1} + 37 \beta_{2} + 14 \beta_{3} + 64 \beta_{4} + 14 \beta_{5} - 62 \beta_{6} + 10 \beta_{7} - 3 \beta_{9} ) q^{73} + ( -30 - 28 \beta_{1} + 140 \beta_{2} + 32 \beta_{3} + 3 \beta_{4} - 18 \beta_{5} - \beta_{6} + 20 \beta_{7} + 38 \beta_{8} - 30 \beta_{9} ) q^{74} + ( -312 - 288 \beta_{2} + 13 \beta_{3} + 16 \beta_{4} - 16 \beta_{6} - 32 \beta_{7} - 32 \beta_{8} + 24 \beta_{9} ) q^{75} + ( -110 + 18 \beta_{1} - 104 \beta_{2} + 17 \beta_{3} + 119 \beta_{4} - 55 \beta_{5} + 53 \beta_{6} + 43 \beta_{7} - 3 \beta_{8} - 54 \beta_{9} ) q^{76} + ( -42 - 8 \beta_{1} + 14 \beta_{2} - 8 \beta_{4} - 12 \beta_{5} - 88 \beta_{6} + 32 \beta_{7} - 32 \beta_{8} - 20 \beta_{9} ) q^{77} + ( -68 + 10 \beta_{1} + 138 \beta_{2} - 56 \beta_{3} + 22 \beta_{4} + 2 \beta_{5} + 86 \beta_{6} + 8 \beta_{7} + 22 \beta_{8} + 4 \beta_{9} ) q^{78} + ( -452 + 12 \beta_{1} + 36 \beta_{2} - 8 \beta_{3} + 40 \beta_{4} + 8 \beta_{5} + 48 \beta_{6} - 56 \beta_{8} + 36 \beta_{9} ) q^{79} + ( -280 + 20 \beta_{1} - 112 \beta_{2} + 74 \beta_{3} + 90 \beta_{4} + 14 \beta_{5} + 34 \beta_{6} - 50 \beta_{7} - 50 \beta_{8} + 8 \beta_{9} ) q^{80} + ( 50 - 15 \beta_{1} - 15 \beta_{2} - 18 \beta_{3} + 6 \beta_{4} + 18 \beta_{5} + 66 \beta_{8} - 15 \beta_{9} ) q^{81} + ( -84 + 4 \beta_{1} - 228 \beta_{2} - 44 \beta_{3} - 12 \beta_{4} + 12 \beta_{5} + 12 \beta_{6} - 60 \beta_{7} + 20 \beta_{8} + 44 \beta_{9} ) q^{82} + ( -240 - 24 \beta_{1} + 256 \beta_{2} - 56 \beta_{4} + 55 \beta_{5} + 104 \beta_{6} - 72 \beta_{7} + 72 \beta_{8} + 40 \beta_{9} ) q^{83} + ( 188 + 8 \beta_{1} + 232 \beta_{2} + 86 \beta_{3} + 42 \beta_{4} - 22 \beta_{5} - 134 \beta_{6} + 26 \beta_{7} - 6 \beta_{8} + 8 \beta_{9} ) q^{84} + ( -22 + 20 \beta_{1} - 18 \beta_{2} - 28 \beta_{3} + 32 \beta_{4} - 112 \beta_{6} + 24 \beta_{7} + 24 \beta_{8} + 24 \beta_{9} ) q^{85} + ( 333 + 3 \beta_{1} + 87 \beta_{2} - 67 \beta_{3} + 112 \beta_{4} + 19 \beta_{5} - 64 \beta_{6} + 29 \beta_{7} + 13 \beta_{8} - 13 \beta_{9} ) q^{86} + ( 17 + 17 \beta_{1} - 547 \beta_{2} + 83 \beta_{3} + 44 \beta_{4} + 83 \beta_{5} + 106 \beta_{6} + 6 \beta_{7} - 39 \beta_{9} ) q^{87} + ( 172 - 22 \beta_{1} - 176 \beta_{2} - 32 \beta_{3} - 60 \beta_{4} - 70 \beta_{5} + 74 \beta_{6} + 4 \beta_{7} - 58 \beta_{8} + 34 \beta_{9} ) q^{88} + ( 53 + 53 \beta_{1} + 35 \beta_{2} - 14 \beta_{3} - 14 \beta_{5} + 110 \beta_{6} + 102 \beta_{7} - 53 \beta_{9} ) q^{89} + ( 172 - 50 \beta_{1} + 290 \beta_{2} - 30 \beta_{3} - 5 \beta_{4} - 72 \beta_{5} - 31 \beta_{6} - 58 \beta_{7} + 4 \beta_{8} + 44 \beta_{9} ) q^{90} + ( 392 + 64 \beta_{1} + 288 \beta_{2} + 70 \beta_{3} - 240 \beta_{4} - 16 \beta_{6} - 32 \beta_{7} - 32 \beta_{8} - 40 \beta_{9} ) q^{91} + ( 98 + 32 \beta_{1} - 434 \beta_{2} + 58 \beta_{3} - 166 \beta_{4} - 4 \beta_{5} - 32 \beta_{6} - 26 \beta_{7} - 28 \beta_{8} + 52 \beta_{9} ) q^{92} + ( -212 - 36 \beta_{1} + 184 \beta_{2} - 100 \beta_{4} - 40 \beta_{5} - 68 \beta_{6} + 12 \beta_{7} - 12 \beta_{8} + 8 \beta_{9} ) q^{93} + ( -372 - 28 \beta_{1} + 252 \beta_{2} + 4 \beta_{3} + 108 \beta_{4} - 4 \beta_{5} - 36 \beta_{6} - 44 \beta_{7} - 28 \beta_{8} - 20 \beta_{9} ) q^{94} + ( 673 - 43 \beta_{1} - 21 \beta_{2} - 107 \beta_{3} - 74 \beta_{4} + 107 \beta_{5} + 44 \beta_{6} + 54 \beta_{8} - 21 \beta_{9} ) q^{95} + ( -432 - 20 \beta_{1} + 304 \beta_{2} - 32 \beta_{3} - 112 \beta_{4} - 108 \beta_{5} - 116 \beta_{6} + 56 \beta_{7} - 52 \beta_{8} - 44 \beta_{9} ) q^{96} + ( 25 - 49 \beta_{1} - 73 \beta_{2} - 10 \beta_{3} - 158 \beta_{4} + 10 \beta_{5} - 48 \beta_{6} + 86 \beta_{8} - 73 \beta_{9} ) q^{97} + ( 36 - 68 \beta_{1} - 364 \beta_{2} - 60 \beta_{3} + 49 \beta_{4} - 100 \beta_{5} - 20 \beta_{6} + 68 \beta_{7} + 52 \beta_{8} + 4 \beta_{9} ) q^{98} + ( 544 + 56 \beta_{1} - 436 \beta_{2} - 12 \beta_{4} + 23 \beta_{5} + 196 \beta_{6} + 20 \beta_{7} - 20 \beta_{8} + 52 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 2q^{2} - 2q^{3} + 8q^{4} - 2q^{5} - 32q^{6} - 44q^{8} + O(q^{10})$$ $$10q - 2q^{2} - 2q^{3} + 8q^{4} - 2q^{5} - 32q^{6} - 44q^{8} - 68q^{10} + 18q^{11} + 100q^{12} - 2q^{13} + 188q^{14} - 124q^{15} + 280q^{16} - 4q^{17} + 174q^{18} - 26q^{19} - 196q^{20} + 52q^{21} - 588q^{22} - 848q^{24} - 264q^{26} + 184q^{27} + 280q^{28} - 202q^{29} + 1236q^{30} + 368q^{31} + 968q^{32} - 4q^{33} + 436q^{34} + 476q^{35} - 596q^{36} - 10q^{37} - 1232q^{38} - 1336q^{40} - 680q^{42} - 838q^{43} + 868q^{44} + 194q^{45} + 1132q^{46} - 944q^{47} + 1768q^{48} + 94q^{49} + 726q^{50} - 1500q^{51} - 236q^{52} - 378q^{53} - 1376q^{54} - 488q^{56} + 8q^{58} + 1706q^{59} - 192q^{60} + 910q^{61} - 80q^{62} + 2628q^{63} + 512q^{64} - 492q^{65} - 428q^{66} + 1942q^{67} - 880q^{68} + 580q^{69} + 160q^{70} + 1092q^{72} - 452q^{74} - 2954q^{75} - 1228q^{76} - 268q^{77} - 772q^{78} - 4416q^{79} - 2648q^{80} + 482q^{81} - 704q^{82} - 2562q^{83} + 1960q^{84} - 12q^{85} + 3764q^{86} + 1528q^{88} + 1896q^{90} + 3332q^{91} + 632q^{92} - 2192q^{93} - 3248q^{94} + 6900q^{95} - 4432q^{96} - 4q^{97} + 314q^{98} + 4958q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} - x^{8} + 6 x^{7} + 14 x^{6} - 80 x^{5} + 56 x^{4} + 96 x^{3} - 64 x^{2} - 512 x + 1024$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{9} - 14 \nu^{8} + 7 \nu^{7} + 82 \nu^{6} - 170 \nu^{5} - 120 \nu^{4} + 536 \nu^{3} + 384 \nu^{2} - 2752 \nu + 3072$$$$)/1280$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{9} - 2 \nu^{8} + 101 \nu^{7} - 114 \nu^{6} + 210 \nu^{5} - 120 \nu^{4} + 8 \nu^{3} - 3008 \nu^{2} + 3264 \nu - 7424$$$$)/1280$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{9} - 18 \nu^{8} + 49 \nu^{7} + 14 \nu^{6} - 230 \nu^{5} + 120 \nu^{4} + 872 \nu^{3} - 1472 \nu^{2} - 1984 \nu + 7424$$$$)/1280$$ $$\beta_{5}$$ $$=$$ $$($$$$-17 \nu^{9} + 38 \nu^{8} - 39 \nu^{7} + 86 \nu^{6} + 90 \nu^{5} + 520 \nu^{4} - 792 \nu^{3} + 2752 \nu^{2} - 1856 \nu - 4864$$$$)/1280$$ $$\beta_{6}$$ $$=$$ $$($$$$17 \nu^{9} - 38 \nu^{8} + 39 \nu^{7} + 74 \nu^{6} - 90 \nu^{5} - 680 \nu^{4} + 1432 \nu^{3} - 512 \nu^{2} - 2624 \nu + 2304$$$$)/1280$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{9} + 6 \nu^{8} + 29 \nu^{7} - 58 \nu^{6} + 18 \nu^{5} + 184 \nu^{4} + 136 \nu^{3} - 1216 \nu^{2} + 1088 \nu - 1280$$$$)/256$$ $$\beta_{8}$$ $$=$$ $$($$$$-37 \nu^{9} + 158 \nu^{8} - 179 \nu^{7} + 46 \nu^{6} - 350 \nu^{5} + 1800 \nu^{4} - 4472 \nu^{3} + 3712 \nu^{2} - 3776 \nu + 5376$$$$)/1280$$ $$\beta_{9}$$ $$=$$ $$($$$$-49 \nu^{9} + 46 \nu^{8} + 297 \nu^{7} - 818 \nu^{6} + 10 \nu^{5} + 2040 \nu^{4} + 616 \nu^{3} - 11136 \nu^{2} + 17088 \nu - 7168$$$$)/1280$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{9} - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{2} + 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} + 6 \beta_{6} + 2 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} + \beta_{1} - 1$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{9} - 2 \beta_{8} - 8 \beta_{6} + 4 \beta_{5} + 14 \beta_{4} - 2 \beta_{3} - 11 \beta_{2} - 34$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-4 \beta_{9} - 4 \beta_{8} + 6 \beta_{7} - 6 \beta_{6} + 6 \beta_{5} + 4 \beta_{4} - 18 \beta_{2} - 5 \beta_{1} + 73$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-15 \beta_{9} - 2 \beta_{8} + 20 \beta_{7} - 28 \beta_{6} + 14 \beta_{4} + 14 \beta_{3} + 27 \beta_{2} + 24 \beta_{1} + 34$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$20 \beta_{9} + 4 \beta_{8} - 46 \beta_{7} + 30 \beta_{6} + 50 \beta_{5} + 12 \beta_{4} + 64 \beta_{3} + 42 \beta_{2} + 5 \beta_{1} + 207$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$7 \beta_{9} + 58 \beta_{8} + 36 \beta_{7} + 292 \beta_{6} + 56 \beta_{5} - 54 \beta_{4} - 22 \beta_{3} - 179 \beta_{2} + 48 \beta_{1} + 326$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$4 \beta_{9} + 28 \beta_{8} - 98 \beta_{7} + 210 \beta_{6} + 94 \beta_{5} + 372 \beta_{4} - 674 \beta_{2} + 35 \beta_{1} - 1111$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$15$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 1.28199 − 1.53509i 1.97476 − 0.316760i −1.62580 − 1.16481i 0.932438 + 1.76934i −1.56339 + 1.24732i 1.28199 + 1.53509i 1.97476 + 0.316760i −1.62580 + 1.16481i 0.932438 − 1.76934i −1.56339 − 1.24732i
−2.81708 0.253099i 5.49618 5.49618i 7.87188 + 1.42600i −4.66372 4.66372i −16.8743 + 14.0921i 24.8965i −21.8148 6.00953i 33.4160i 11.9577 + 14.3185i
5.2 −2.29152 + 1.65800i −5.96513 + 5.96513i 2.50210 7.59865i 8.67959 + 8.67959i 3.77903 23.5594i 1.63924i 6.86495 + 21.5609i 44.1656i −34.2802 5.49869i
5.3 0.460984 2.79061i 0.756776 0.756776i −7.57499 2.57285i 8.22587 + 8.22587i −1.76300 2.46073i 2.67171i −10.6718 + 19.9528i 25.8546i 26.7472 19.1632i
5.4 0.836901 + 2.70178i 1.98356 1.98356i −6.59919 + 4.52224i −0.596848 0.596848i 7.01918 + 3.69910i 29.0828i −17.7410 14.0449i 19.1310i 1.11305 2.11205i
5.5 2.81071 0.316066i −3.27139 + 3.27139i 7.80020 1.77674i −12.6449 12.6449i −8.16095 + 10.2289i 13.8754i 21.3626 7.45928i 5.59607i −39.5378 31.5445i
13.1 −2.81708 + 0.253099i 5.49618 + 5.49618i 7.87188 1.42600i −4.66372 + 4.66372i −16.8743 14.0921i 24.8965i −21.8148 + 6.00953i 33.4160i 11.9577 14.3185i
13.2 −2.29152 1.65800i −5.96513 5.96513i 2.50210 + 7.59865i 8.67959 8.67959i 3.77903 + 23.5594i 1.63924i 6.86495 21.5609i 44.1656i −34.2802 + 5.49869i
13.3 0.460984 + 2.79061i 0.756776 + 0.756776i −7.57499 + 2.57285i 8.22587 8.22587i −1.76300 + 2.46073i 2.67171i −10.6718 19.9528i 25.8546i 26.7472 + 19.1632i
13.4 0.836901 2.70178i 1.98356 + 1.98356i −6.59919 4.52224i −0.596848 + 0.596848i 7.01918 3.69910i 29.0828i −17.7410 + 14.0449i 19.1310i 1.11305 + 2.11205i
13.5 2.81071 + 0.316066i −3.27139 3.27139i 7.80020 + 1.77674i −12.6449 + 12.6449i −8.16095 10.2289i 13.8754i 21.3626 + 7.45928i 5.59607i −39.5378 + 31.5445i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.4.e.a 10
3.b odd 2 1 144.4.k.a 10
4.b odd 2 1 64.4.e.a 10
8.b even 2 1 128.4.e.b 10
8.d odd 2 1 128.4.e.a 10
12.b even 2 1 576.4.k.a 10
16.e even 4 1 inner 16.4.e.a 10
16.e even 4 1 128.4.e.b 10
16.f odd 4 1 64.4.e.a 10
16.f odd 4 1 128.4.e.a 10
32.g even 8 2 1024.4.a.n 10
32.g even 8 2 1024.4.b.j 10
32.h odd 8 2 1024.4.a.m 10
32.h odd 8 2 1024.4.b.k 10
48.i odd 4 1 144.4.k.a 10
48.k even 4 1 576.4.k.a 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.4.e.a 10 1.a even 1 1 trivial
16.4.e.a 10 16.e even 4 1 inner
64.4.e.a 10 4.b odd 2 1
64.4.e.a 10 16.f odd 4 1
128.4.e.a 10 8.d odd 2 1
128.4.e.a 10 16.f odd 4 1
128.4.e.b 10 8.b even 2 1
128.4.e.b 10 16.e even 4 1
144.4.k.a 10 3.b odd 2 1
144.4.k.a 10 48.i odd 4 1
576.4.k.a 10 12.b even 2 1
576.4.k.a 10 48.k even 4 1
1024.4.a.m 10 32.h odd 8 2
1024.4.a.n 10 32.g even 8 2
1024.4.b.j 10 32.g even 8 2
1024.4.b.k 10 32.h odd 8 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(16, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T - 2 T^{2} + 8 T^{3} - 40 T^{4} - 352 T^{5} - 320 T^{6} + 512 T^{7} - 1024 T^{8} + 8192 T^{9} + 32768 T^{10}$$
$3$ $$1 + 2 T + 2 T^{2} - 42 T^{3} - 571 T^{4} + 760 T^{5} + 3544 T^{6} + 66792 T^{7} + 57874 T^{8} - 3046228 T^{9} - 6701044 T^{10} - 82248156 T^{11} + 42190146 T^{12} + 1314666936 T^{13} + 1883426904 T^{14} + 10905169320 T^{15} - 221217099219 T^{16} - 439334834526 T^{17} + 564859072962 T^{18} + 15251194969974 T^{19} + 205891132094649 T^{20}$$
$5$ $$1 + 2 T + 2 T^{2} - 966 T^{3} - 13723 T^{4} + 18040 T^{5} + 530104 T^{6} - 11981288 T^{7} - 28535006 T^{8} + 2301854348 T^{9} + 23672040908 T^{10} + 287731793500 T^{11} - 445859468750 T^{12} - 23400953125000 T^{13} + 129419921875000 T^{14} + 550537109375000 T^{15} - 52349090576171875 T^{16} - 460624694824218750 T^{17} + 119209289550781250 T^{18} + 14901161193847656250 T^{19} +$$$$93\!\cdots\!25$$$$T^{20}$$
$7$ $$1 - 1762 T^{2} + 1539965 T^{4} - 932087576 T^{6} + 440869947922 T^{8} - 168121217547916 T^{10} + 51867908503075378 T^{12} - 12901291835899914776 T^{14} +$$$$25\!\cdots\!85$$$$T^{16} -$$$$33\!\cdots\!62$$$$T^{18} +$$$$22\!\cdots\!49$$$$T^{20}$$
$11$ $$1 - 18 T + 162 T^{2} - 122934 T^{3} + 4077397 T^{4} + 79597000 T^{5} + 5463099864 T^{6} - 313798751208 T^{7} - 6887886337838 T^{8} + 101615185776500 T^{9} + 18755914132083020 T^{10} + 135249812268521500 T^{11} - 12202310808546625118 T^{12} -$$$$73\!\cdots\!28$$$$T^{13} +$$$$17\!\cdots\!44$$$$T^{14} +$$$$33\!\cdots\!00$$$$T^{15} +$$$$22\!\cdots\!57$$$$T^{16} -$$$$90\!\cdots\!74$$$$T^{17} +$$$$15\!\cdots\!42$$$$T^{18} -$$$$23\!\cdots\!78$$$$T^{19} +$$$$17\!\cdots\!01$$$$T^{20}$$
$13$ $$1 + 2 T + 2 T^{2} - 45206 T^{3} - 2401451 T^{4} + 29395960 T^{5} + 1085386040 T^{6} - 384228102440 T^{7} - 9400034983966 T^{8} + 1531455561616908 T^{9} + 23489689415409228 T^{10} + 3364607868872346876 T^{11} - 45372173460921944494 T^{12} -$$$$40\!\cdots\!20$$$$T^{13} +$$$$25\!\cdots\!40$$$$T^{14} +$$$$15\!\cdots\!20$$$$T^{15} -$$$$27\!\cdots\!79$$$$T^{16} -$$$$11\!\cdots\!78$$$$T^{17} +$$$$10\!\cdots\!22$$$$T^{18} +$$$$23\!\cdots\!34$$$$T^{19} +$$$$26\!\cdots\!49$$$$T^{20}$$
$17$ $$( 1 + 2 T + 12653 T^{2} + 102520 T^{3} + 98460610 T^{4} + 354493580 T^{5} + 483736976930 T^{6} + 2474583573880 T^{7} + 1500492401316541 T^{8} + 1165244474459522 T^{9} + 2862423051509815793 T^{10} )^{2}$$
$19$ $$1 + 26 T + 338 T^{2} - 339906 T^{3} - 64153371 T^{4} + 2461461784 T^{5} + 143449890200 T^{6} + 36783398837960 T^{7} + 1011857007777554 T^{8} - 259766590630759364 T^{9} - 7366645907488092948 T^{10} -$$$$17\!\cdots\!76$$$$T^{11} +$$$$47\!\cdots\!74$$$$T^{12} +$$$$11\!\cdots\!40$$$$T^{13} +$$$$31\!\cdots\!00$$$$T^{14} +$$$$37\!\cdots\!16$$$$T^{15} -$$$$66\!\cdots\!11$$$$T^{16} -$$$$24\!\cdots\!14$$$$T^{17} +$$$$16\!\cdots\!98$$$$T^{18} +$$$$87\!\cdots\!14$$$$T^{19} +$$$$23\!\cdots\!01$$$$T^{20}$$
$23$ $$1 - 76386 T^{2} + 2913757597 T^{4} - 73253961622040 T^{6} + 1342371312768300946 T^{8} -$$$$18\!\cdots\!84$$$$T^{10} +$$$$19\!\cdots\!94$$$$T^{12} -$$$$16\!\cdots\!40$$$$T^{14} +$$$$94\!\cdots\!93$$$$T^{16} -$$$$36\!\cdots\!26$$$$T^{18} +$$$$71\!\cdots\!49$$$$T^{20}$$
$29$ $$1 + 202 T + 20402 T^{2} - 1177934 T^{3} + 398569397 T^{4} + 239164019416 T^{5} + 40873283338616 T^{6} + 2529271278095288 T^{7} + 194871598558001506 T^{8} +$$$$12\!\cdots\!32$$$$T^{9} +$$$$30\!\cdots\!36$$$$T^{10} +$$$$30\!\cdots\!48$$$$T^{11} +$$$$11\!\cdots\!26$$$$T^{12} +$$$$36\!\cdots\!72$$$$T^{13} +$$$$14\!\cdots\!56$$$$T^{14} +$$$$20\!\cdots\!84$$$$T^{15} +$$$$83\!\cdots\!17$$$$T^{16} -$$$$60\!\cdots\!86$$$$T^{17} +$$$$25\!\cdots\!62$$$$T^{18} +$$$$61\!\cdots\!18$$$$T^{19} +$$$$74\!\cdots\!01$$$$T^{20}$$
$31$ $$( 1 - 184 T + 134043 T^{2} - 19809056 T^{3} + 7638677322 T^{4} - 852982867024 T^{5} + 227563836099702 T^{6} - 17580610117135136 T^{7} + 3544046273282822853 T^{8} -$$$$14\!\cdots\!24$$$$T^{9} +$$$$23\!\cdots\!51$$$$T^{10} )^{2}$$
$37$ $$1 + 10 T + 50 T^{2} + 1972962 T^{3} + 1630465317 T^{4} - 153991562664 T^{5} + 324850634232 T^{6} - 5252842710654600 T^{7} + 3474549392106364962 T^{8} + 27985624577691139772 T^{9} +$$$$10\!\cdots\!24$$$$T^{10} +$$$$14\!\cdots\!16$$$$T^{11} +$$$$89\!\cdots\!58$$$$T^{12} -$$$$68\!\cdots\!00$$$$T^{13} +$$$$21\!\cdots\!92$$$$T^{14} -$$$$51\!\cdots\!52$$$$T^{15} +$$$$27\!\cdots\!93$$$$T^{16} +$$$$16\!\cdots\!94$$$$T^{17} +$$$$21\!\cdots\!50$$$$T^{18} +$$$$21\!\cdots\!30$$$$T^{19} +$$$$11\!\cdots\!49$$$$T^{20}$$
$41$ $$1 - 441018 T^{2} + 97166156061 T^{4} - 13934678680622904 T^{6} +$$$$14\!\cdots\!14$$$$T^{8} -$$$$11\!\cdots\!88$$$$T^{10} +$$$$68\!\cdots\!74$$$$T^{12} -$$$$31\!\cdots\!24$$$$T^{14} +$$$$10\!\cdots\!81$$$$T^{16} -$$$$22\!\cdots\!98$$$$T^{18} +$$$$24\!\cdots\!01$$$$T^{20}$$
$43$ $$1 + 838 T + 351122 T^{2} + 132133650 T^{3} + 56398378005 T^{4} + 20936462157416 T^{5} + 6471694737204248 T^{6} + 1952595983380873720 T^{7} +$$$$59\!\cdots\!10$$$$T^{8} +$$$$17\!\cdots\!68$$$$T^{9} +$$$$49\!\cdots\!52$$$$T^{10} +$$$$13\!\cdots\!76$$$$T^{11} +$$$$37\!\cdots\!90$$$$T^{12} +$$$$98\!\cdots\!60$$$$T^{13} +$$$$25\!\cdots\!48$$$$T^{14} +$$$$66\!\cdots\!12$$$$T^{15} +$$$$14\!\cdots\!45$$$$T^{16} +$$$$26\!\cdots\!50$$$$T^{17} +$$$$56\!\cdots\!22$$$$T^{18} +$$$$10\!\cdots\!66$$$$T^{19} +$$$$10\!\cdots\!49$$$$T^{20}$$
$47$ $$( 1 + 472 T + 462219 T^{2} + 171516064 T^{3} + 90105579914 T^{4} + 25593405310224 T^{5} + 9355031623411222 T^{6} + 1848808586238545056 T^{7} +$$$$51\!\cdots\!73$$$$T^{8} +$$$$54\!\cdots\!52$$$$T^{9} +$$$$12\!\cdots\!43$$$$T^{10} )^{2}$$
$53$ $$1 + 378 T + 71442 T^{2} - 52753550 T^{3} + 3016286341 T^{4} - 526686651752 T^{5} + 976891435665272 T^{6} - 1674349213754452168 T^{7} - 1581505854923305054 T^{8} -$$$$14\!\cdots\!20$$$$T^{9} +$$$$29\!\cdots\!08$$$$T^{10} -$$$$22\!\cdots\!40$$$$T^{11} -$$$$35\!\cdots\!66$$$$T^{12} -$$$$55\!\cdots\!44$$$$T^{13} +$$$$47\!\cdots\!52$$$$T^{14} -$$$$38\!\cdots\!64$$$$T^{15} +$$$$32\!\cdots\!49$$$$T^{16} -$$$$85\!\cdots\!50$$$$T^{17} +$$$$17\!\cdots\!02$$$$T^{18} +$$$$13\!\cdots\!86$$$$T^{19} +$$$$53\!\cdots\!49$$$$T^{20}$$
$59$ $$1 - 1706 T + 1455218 T^{2} - 989315358 T^{3} + 555128806581 T^{4} - 232658117164632 T^{5} + 78453755015006616 T^{6} - 20092577710244830152 T^{7} +$$$$57\!\cdots\!98$$$$T^{8} +$$$$22\!\cdots\!08$$$$T^{9} -$$$$12\!\cdots\!80$$$$T^{10} +$$$$45\!\cdots\!32$$$$T^{11} +$$$$24\!\cdots\!18$$$$T^{12} -$$$$17\!\cdots\!28$$$$T^{13} +$$$$13\!\cdots\!96$$$$T^{14} -$$$$85\!\cdots\!68$$$$T^{15} +$$$$41\!\cdots\!01$$$$T^{16} -$$$$15\!\cdots\!22$$$$T^{17} +$$$$46\!\cdots\!98$$$$T^{18} -$$$$11\!\cdots\!14$$$$T^{19} +$$$$13\!\cdots\!01$$$$T^{20}$$
$61$ $$1 - 910 T + 414050 T^{2} + 45940410 T^{3} + 20471098485 T^{4} - 72823214590920 T^{5} + 58848327585507000 T^{6} - 6001380717052735080 T^{7} +$$$$59\!\cdots\!50$$$$T^{8} -$$$$32\!\cdots\!00$$$$T^{9} +$$$$38\!\cdots\!00$$$$T^{10} -$$$$73\!\cdots\!00$$$$T^{11} +$$$$30\!\cdots\!50$$$$T^{12} -$$$$70\!\cdots\!80$$$$T^{13} +$$$$15\!\cdots\!00$$$$T^{14} -$$$$43\!\cdots\!20$$$$T^{15} +$$$$27\!\cdots\!85$$$$T^{16} +$$$$14\!\cdots\!10$$$$T^{17} +$$$$29\!\cdots\!50$$$$T^{18} -$$$$14\!\cdots\!10$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20}$$
$67$ $$1 - 1942 T + 1885682 T^{2} - 1530965298 T^{3} + 1338066168261 T^{4} - 1068751594464168 T^{5} + 724275679990789656 T^{6} -$$$$46\!\cdots\!12$$$$T^{7} +$$$$29\!\cdots\!06$$$$T^{8} -$$$$17\!\cdots\!64$$$$T^{9} +$$$$97\!\cdots\!68$$$$T^{10} -$$$$52\!\cdots\!32$$$$T^{11} +$$$$26\!\cdots\!14$$$$T^{12} -$$$$12\!\cdots\!64$$$$T^{13} +$$$$59\!\cdots\!16$$$$T^{14} -$$$$26\!\cdots\!24$$$$T^{15} +$$$$99\!\cdots\!49$$$$T^{16} -$$$$34\!\cdots\!66$$$$T^{17} +$$$$12\!\cdots\!22$$$$T^{18} -$$$$39\!\cdots\!66$$$$T^{19} +$$$$60\!\cdots\!49$$$$T^{20}$$
$71$ $$1 - 2500418 T^{2} + 3072516920573 T^{4} - 2420243241413642648 T^{6} +$$$$13\!\cdots\!66$$$$T^{8} -$$$$55\!\cdots\!32$$$$T^{10} +$$$$17\!\cdots\!86$$$$T^{12} -$$$$39\!\cdots\!68$$$$T^{14} +$$$$64\!\cdots\!53$$$$T^{16} -$$$$67\!\cdots\!58$$$$T^{18} +$$$$34\!\cdots\!01$$$$T^{20}$$
$73$ $$1 - 3134282 T^{2} + 4627160201821 T^{4} - 4242124717558516472 T^{6} +$$$$26\!\cdots\!74$$$$T^{8} -$$$$12\!\cdots\!92$$$$T^{10} +$$$$40\!\cdots\!86$$$$T^{12} -$$$$97\!\cdots\!12$$$$T^{14} +$$$$16\!\cdots\!49$$$$T^{16} -$$$$16\!\cdots\!62$$$$T^{18} +$$$$79\!\cdots\!49$$$$T^{20}$$
$79$ $$( 1 + 2208 T + 3816107 T^{2} + 4320867712 T^{3} + 4245684014154 T^{4} + 3176789940661184 T^{5} + 2093287800654474006 T^{6} +$$$$10\!\cdots\!52$$$$T^{7} +$$$$45\!\cdots\!33$$$$T^{8} +$$$$13\!\cdots\!28$$$$T^{9} +$$$$29\!\cdots\!99$$$$T^{10} )^{2}$$
$83$ $$1 + 2562 T + 3281922 T^{2} + 2891460918 T^{3} + 1934799974629 T^{4} + 1191348439341176 T^{5} + 882645219418437336 T^{6} +$$$$79\!\cdots\!28$$$$T^{7} +$$$$89\!\cdots\!74$$$$T^{8} +$$$$97\!\cdots\!88$$$$T^{9} +$$$$82\!\cdots\!76$$$$T^{10} +$$$$55\!\cdots\!56$$$$T^{11} +$$$$29\!\cdots\!06$$$$T^{12} +$$$$14\!\cdots\!84$$$$T^{13} +$$$$94\!\cdots\!96$$$$T^{14} +$$$$72\!\cdots\!32$$$$T^{15} +$$$$67\!\cdots\!61$$$$T^{16} +$$$$57\!\cdots\!94$$$$T^{17} +$$$$37\!\cdots\!62$$$$T^{18} +$$$$16\!\cdots\!74$$$$T^{19} +$$$$37\!\cdots\!49$$$$T^{20}$$
$89$ $$1 - 3643178 T^{2} + 6505439011133 T^{4} - 7720859292932177528 T^{6} +$$$$69\!\cdots\!98$$$$T^{8} -$$$$52\!\cdots\!28$$$$T^{10} +$$$$34\!\cdots\!78$$$$T^{12} -$$$$19\!\cdots\!88$$$$T^{14} +$$$$79\!\cdots\!73$$$$T^{16} -$$$$22\!\cdots\!98$$$$T^{18} +$$$$30\!\cdots\!01$$$$T^{20}$$
$97$ $$( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 2948979193634928834 T^{6} -$$$$60\!\cdots\!36$$$$T^{7} +$$$$19\!\cdots\!13$$$$T^{8} +$$$$13\!\cdots\!82$$$$T^{9} +$$$$63\!\cdots\!93$$$$T^{10} )^{2}$$