# Properties

 Label 38.4.e.a Level $38$ Weight $4$ Character orbit 38.e Analytic conductor $2.242$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 38.e (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.24207258022$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 6 x^{11} - 135 x^{10} + 730 x^{9} + 7953 x^{8} - 36258 x^{7} - 262940 x^{6} + 918855 x^{5} + 5157591 x^{4} - 11890401 x^{3} - 56759508 x^{2} + 62864118 x + 272110107$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \beta_{3} q^{2} + ( -1 + 2 \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{3} -4 \beta_{6} q^{4} + ( 1 - 2 \beta_{1} - 3 \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{8} + \beta_{9} + \beta_{10} ) q^{5} + ( -2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} ) q^{6} + ( -\beta_{1} + 6 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} - 4 \beta_{7} - 7 \beta_{8} + \beta_{9} ) q^{7} + ( -8 - 8 \beta_{7} ) q^{8} + ( -3 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{7} + 3 \beta_{10} + 3 \beta_{11} ) q^{9} +O(q^{10})$$ $$q -2 \beta_{3} q^{2} + ( -1 + 2 \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{3} -4 \beta_{6} q^{4} + ( 1 - 2 \beta_{1} - 3 \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{8} + \beta_{9} + \beta_{10} ) q^{5} + ( -2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} ) q^{6} + ( -\beta_{1} + 6 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} - 4 \beta_{7} - 7 \beta_{8} + \beta_{9} ) q^{7} + ( -8 - 8 \beta_{7} ) q^{8} + ( -3 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{7} + 3 \beta_{10} + 3 \beta_{11} ) q^{9} + ( 2 + 4 \beta_{1} + 12 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 4 \beta_{11} ) q^{10} + ( -6 - 4 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 16 \beta_{6} - 4 \beta_{7} - 13 \beta_{8} - 2 \beta_{10} + 5 \beta_{11} ) q^{11} + ( 4 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} + 4 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} ) q^{12} + ( 9 + \beta_{1} - 5 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} + 10 \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{13} + ( -8 - 12 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} - 10 \beta_{6} + 4 \beta_{7} + 6 \beta_{8} - 2 \beta_{11} ) q^{14} + ( 44 + 3 \beta_{1} + 28 \beta_{3} + 3 \beta_{4} + 9 \beta_{5} + 3 \beta_{6} + 34 \beta_{7} + 19 \beta_{8} + 3 \beta_{9} - 9 \beta_{10} - 3 \beta_{11} ) q^{15} + 16 \beta_{8} q^{16} + ( -6 - 6 \beta_{1} - 33 \beta_{2} + 22 \beta_{3} - 6 \beta_{5} - 21 \beta_{6} - 33 \beta_{7} + 7 \beta_{9} - \beta_{10} - 5 \beta_{11} ) q^{17} + ( 8 + 6 \beta_{1} + 6 \beta_{3} - 12 \beta_{6} - 6 \beta_{8} - 6 \beta_{10} - 6 \beta_{11} ) q^{18} + ( -29 + 3 \beta_{1} + 18 \beta_{2} - 7 \beta_{4} - 13 \beta_{5} + 9 \beta_{6} + 10 \beta_{7} + 2 \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{19} + ( -20 + 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{7} + 8 \beta_{10} + 8 \beta_{11} ) q^{20} + ( -23 - 5 \beta_{2} + 10 \beta_{3} + 18 \beta_{6} - 5 \beta_{7} - 5 \beta_{9} + 5 \beta_{10} + 2 \beta_{11} ) q^{21} + ( 30 - 10 \beta_{1} - 40 \beta_{2} + 6 \beta_{5} - 26 \beta_{6} + 32 \beta_{7} + 2 \beta_{8} + 6 \beta_{9} - 10 \beta_{10} ) q^{22} + ( -23 + 13 \beta_{1} - 17 \beta_{3} + 13 \beta_{4} + \beta_{5} + 38 \beta_{6} - 12 \beta_{7} + 7 \beta_{8} - 4 \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{23} + ( -8 - 16 \beta_{2} - 16 \beta_{3} + 8 \beta_{4} + 8 \beta_{7} + 16 \beta_{8} + 8 \beta_{11} ) q^{24} + ( 18 + 9 \beta_{1} + 43 \beta_{2} - 18 \beta_{4} + 25 \beta_{6} - 39 \beta_{7} - 57 \beta_{8} - 18 \beta_{9} - 9 \beta_{11} ) q^{25} + ( -2 - 10 \beta_{1} + 10 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} - 14 \beta_{7} - 20 \beta_{8} + 10 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{26} + ( 31 + 9 \beta_{4} + 18 \beta_{5} - 49 \beta_{6} + 40 \beta_{7} + 49 \beta_{8} - 8 \beta_{9} - 9 \beta_{10} - 9 \beta_{11} ) q^{27} + ( 8 + 16 \beta_{2} + 28 \beta_{3} - 20 \beta_{7} - 8 \beta_{8} + 4 \beta_{10} ) q^{28} + ( 39 - \beta_{1} + 91 \beta_{2} + 13 \beta_{3} + 4 \beta_{4} - 3 \beta_{6} + 26 \beta_{7} - 36 \beta_{8} - 3 \beta_{9} + 26 \beta_{10} + \beta_{11} ) q^{29} + ( 12 + 38 \beta_{2} - 38 \beta_{3} - 6 \beta_{4} - 12 \beta_{5} + 94 \beta_{6} + 6 \beta_{7} - 56 \beta_{8} - 18 \beta_{9} + 6 \beta_{10} + 6 \beta_{11} ) q^{30} + ( -13 - 9 \beta_{1} - 15 \beta_{2} - 72 \beta_{3} - 12 \beta_{4} + 13 \beta_{5} - 34 \beta_{6} - 4 \beta_{7} + 6 \beta_{8} + 9 \beta_{9} - 25 \beta_{10} + 13 \beta_{11} ) q^{31} + ( 32 \beta_{2} + 32 \beta_{6} ) q^{32} + ( -141 - 25 \beta_{1} - 111 \beta_{2} - 43 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} - 138 \beta_{6} - 51 \beta_{7} + 43 \beta_{8} + 21 \beta_{9} + 21 \beta_{10} + 3 \beta_{11} ) q^{33} + ( 34 - 14 \beta_{1} - 20 \beta_{3} - 14 \beta_{4} - 10 \beta_{5} + 36 \beta_{6} - 42 \beta_{7} + 40 \beta_{8} + 12 \beta_{9} + 10 \beta_{10} - 12 \beta_{11} ) q^{34} + ( -34 + 19 \beta_{2} + 15 \beta_{4} + 15 \beta_{5} + 23 \beta_{6} - 23 \beta_{7} + 15 \beta_{8} + 15 \beta_{9} ) q^{35} + ( -12 - 12 \beta_{1} - 24 \beta_{2} - 16 \beta_{3} - 12 \beta_{5} - 24 \beta_{7} + 12 \beta_{10} + 12 \beta_{11} ) q^{36} + ( 96 + 21 \beta_{1} - \beta_{2} + 11 \beta_{3} - 27 \beta_{4} + 27 \beta_{5} - 33 \beta_{6} + 27 \beta_{7} - 68 \beta_{8} - 19 \beta_{10} - 19 \beta_{11} ) q^{37} + ( -20 - 18 \beta_{1} + 96 \beta_{3} - 4 \beta_{4} + 16 \beta_{5} + 26 \beta_{6} + 18 \beta_{7} - 50 \beta_{8} + 26 \beta_{9} - 2 \beta_{10} + 6 \beta_{11} ) q^{38} + ( -98 - 8 \beta_{1} + 17 \beta_{2} - \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 26 \beta_{6} + 4 \beta_{7} + 12 \beta_{8} + 6 \beta_{10} + 6 \beta_{11} ) q^{39} + ( -16 + 8 \beta_{1} + 24 \beta_{3} + 8 \beta_{5} + 8 \beta_{6} + 8 \beta_{9} - 16 \beta_{10} ) q^{40} + ( 82 + 7 \beta_{1} - 72 \beta_{2} - 3 \beta_{4} - 21 \beta_{5} - 135 \beta_{6} + 117 \beta_{7} - 50 \beta_{8} - 21 \beta_{9} + 7 \beta_{10} ) q^{41} + ( 42 + 10 \beta_{1} + 32 \beta_{3} + 10 \beta_{4} + 4 \beta_{5} + 14 \beta_{6} + 36 \beta_{7} + 20 \beta_{8} - 4 \beta_{10} ) q^{42} + ( 25 + 4 \beta_{1} - 36 \beta_{2} - 337 \beta_{3} - 7 \beta_{4} - 15 \beta_{5} + 18 \beta_{6} + 50 \beta_{7} + 337 \beta_{8} + 11 \beta_{9} + 11 \beta_{10} - 7 \beta_{11} ) q^{43} + ( 28 - 8 \beta_{1} + 8 \beta_{2} + 12 \beta_{3} - 12 \beta_{4} - 4 \beta_{6} - 52 \beta_{7} - 64 \beta_{8} - 12 \beta_{9} - 20 \beta_{11} ) q^{44} + ( 28 + 27 \beta_{1} + 282 \beta_{2} + 114 \beta_{3} - 29 \beta_{4} - 28 \beta_{5} + 87 \beta_{6} - 202 \beta_{7} - 255 \beta_{8} - 27 \beta_{9} - \beta_{10} - 28 \beta_{11} ) q^{45} + ( 68 - 20 \beta_{2} - 14 \beta_{3} + 8 \beta_{4} - 18 \beta_{5} - 20 \beta_{6} + 76 \beta_{7} + 34 \beta_{8} - 2 \beta_{9} - 8 \beta_{10} + 26 \beta_{11} ) q^{46} + ( -29 - 5 \beta_{1} - 178 \beta_{2} + 126 \beta_{3} + 34 \beta_{4} - 29 \beta_{6} - 155 \beta_{7} + 58 \beta_{8} - 29 \beta_{9} - 31 \beta_{10} + 5 \beta_{11} ) q^{47} + ( 16 + 16 \beta_{2} + 16 \beta_{3} - 16 \beta_{8} - 16 \beta_{10} ) q^{48} + ( 234 + 18 \beta_{2} - 11 \beta_{3} + 3 \beta_{4} + 13 \beta_{5} + 64 \beta_{6} + 237 \beta_{7} - 53 \beta_{8} + 7 \beta_{9} - 3 \beta_{10} - 10 \beta_{11} ) q^{49} + ( -18 - 114 \beta_{2} - 114 \beta_{3} + 36 \beta_{4} + 18 \beta_{5} - 114 \beta_{6} + 50 \beta_{7} + 114 \beta_{8} + 18 \beta_{10} + 18 \beta_{11} ) q^{50} + ( -215 + 28 \beta_{1} + 219 \beta_{2} + 165 \beta_{3} - 3 \beta_{4} - 17 \beta_{5} + 216 \beta_{6} - 5 \beta_{7} + 9 \beta_{8} - 3 \beta_{9} + 25 \beta_{11} ) q^{51} + ( -12 + 8 \beta_{1} - 20 \beta_{2} - 12 \beta_{3} - 20 \beta_{4} - 4 \beta_{5} - 32 \beta_{6} + 12 \beta_{7} + 12 \beta_{8} - 4 \beta_{9} - 4 \beta_{10} - 20 \beta_{11} ) q^{52} + ( 6 + 43 \beta_{1} + 92 \beta_{3} + 43 \beta_{4} - 19 \beta_{5} + 96 \beta_{6} + 106 \beta_{7} - 43 \beta_{8} - 33 \beta_{9} + 19 \beta_{10} + 33 \beta_{11} ) q^{53} + ( -80 + 18 \beta_{1} + 82 \beta_{2} + 16 \beta_{4} - 36 \beta_{5} + 46 \beta_{6} - 98 \beta_{7} - 28 \beta_{8} - 36 \beta_{9} + 18 \beta_{10} ) q^{54} + ( 66 + 8 \beta_{1} - 199 \beta_{2} + 180 \beta_{3} + 8 \beta_{5} - 273 \beta_{6} - 199 \beta_{7} + 41 \beta_{9} - 49 \beta_{10} - 48 \beta_{11} ) q^{55} + ( -32 + 8 \beta_{1} - 8 \beta_{2} - 56 \beta_{3} + 40 \beta_{6} + 40 \beta_{8} ) q^{56} + ( -132 - 39 \beta_{1} - 266 \beta_{2} + 132 \beta_{3} + 42 \beta_{4} + 3 \beta_{5} - 64 \beta_{6} + 153 \beta_{7} - 102 \beta_{8} - 7 \beta_{9} + 21 \beta_{10} + 13 \beta_{11} ) q^{57} + ( -190 + 52 \beta_{1} - 34 \beta_{2} - 26 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} - 60 \beta_{6} - 6 \beta_{7} - 46 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{58} + ( -385 - 12 \beta_{1} - 15 \beta_{2} + 48 \beta_{3} - 12 \beta_{5} + 382 \beta_{6} - 15 \beta_{7} + 36 \beta_{9} - 24 \beta_{10} + 23 \beta_{11} ) q^{59} + ( 100 - 12 \beta_{1} - 148 \beta_{2} + 36 \beta_{4} + 24 \beta_{5} - 200 \beta_{6} + 188 \beta_{7} + 24 \beta_{9} - 12 \beta_{10} ) q^{60} + ( 271 - 46 \beta_{1} - 81 \beta_{3} - 46 \beta_{4} - 61 \beta_{5} + 97 \beta_{6} - 133 \beta_{7} + 306 \beta_{8} - 9 \beta_{9} + 61 \beta_{10} + 9 \beta_{11} ) q^{61} + ( -64 - 24 \beta_{1} + 30 \beta_{2} - 16 \beta_{3} - 18 \beta_{4} + 50 \beta_{5} - 82 \beta_{6} - 68 \beta_{7} + 16 \beta_{8} - 26 \beta_{9} - 26 \beta_{10} - 18 \beta_{11} ) q^{62} + ( -10 - 30 \beta_{1} - 37 \beta_{2} - 20 \beta_{3} + 45 \beta_{4} + 29 \beta_{5} + 8 \beta_{6} - 36 \beta_{7} - 20 \beta_{8} + 45 \beta_{9} + 15 \beta_{11} ) q^{63} + 64 \beta_{7} q^{64} + ( 100 - 53 \beta_{2} + 74 \beta_{3} - 14 \beta_{4} - 7 \beta_{5} - 247 \beta_{6} + 86 \beta_{7} + 173 \beta_{8} + 14 \beta_{10} - 7 \beta_{11} ) q^{65} + ( -60 + 50 \beta_{1} + 172 \beta_{2} + 216 \beta_{3} - 42 \beta_{4} - 8 \beta_{6} - 276 \beta_{7} + 68 \beta_{8} - 8 \beta_{9} - 6 \beta_{10} - 50 \beta_{11} ) q^{66} + ( 298 - 24 \beta_{1} + 315 \beta_{2} + 183 \beta_{3} + 5 \beta_{4} + 19 \beta_{6} + 115 \beta_{7} - 317 \beta_{8} + 19 \beta_{9} - 17 \beta_{10} + 24 \beta_{11} ) q^{67} + ( 96 + 132 \beta_{2} - 80 \beta_{3} - 24 \beta_{4} + 4 \beta_{5} + 40 \beta_{6} + 72 \beta_{7} + 40 \beta_{8} + 20 \beta_{9} + 24 \beta_{10} - 28 \beta_{11} ) q^{68} + ( 7 - 35 \beta_{1} + 68 \beta_{2} - 211 \beta_{3} - 19 \beta_{4} - 7 \beta_{5} - 206 \beta_{6} + 28 \beta_{7} - 103 \beta_{8} + 35 \beta_{9} - 12 \beta_{10} - 7 \beta_{11} ) q^{69} + ( 8 + 30 \beta_{1} + 60 \beta_{2} - 8 \beta_{3} - 30 \beta_{4} - 30 \beta_{5} + 30 \beta_{6} + 46 \beta_{7} + 46 \beta_{8} - 30 \beta_{9} ) q^{70} + ( 145 + 47 \beta_{1} + 165 \beta_{2} - 254 \beta_{3} - 35 \beta_{4} + 21 \beta_{5} + 110 \beta_{6} + 48 \beta_{7} + 254 \beta_{8} - 68 \beta_{9} - 68 \beta_{10} - 35 \beta_{11} ) q^{71} + ( 24 + 24 \beta_{5} - 32 \beta_{6} + 24 \beta_{8} + 24 \beta_{9} - 24 \beta_{10} - 24 \beta_{11} ) q^{72} + ( 73 - 18 \beta_{1} - 81 \beta_{2} - 10 \beta_{4} + 29 \beta_{5} + 140 \beta_{6} - 101 \beta_{7} - 473 \beta_{8} + 29 \beta_{9} - 18 \beta_{10} ) q^{73} + ( -26 + 16 \beta_{1} - 66 \beta_{2} - 196 \beta_{3} + 16 \beta_{5} - 56 \beta_{6} - 66 \beta_{7} - 54 \beta_{9} + 38 \beta_{10} + 42 \beta_{11} ) q^{74} + ( -133 - 70 \beta_{1} - 368 \beta_{2} - 583 \beta_{3} + 27 \beta_{4} - 27 \beta_{5} + 285 \beta_{6} - 27 \beta_{7} + 228 \beta_{8} + 111 \beta_{10} + 111 \beta_{11} ) q^{75} + ( 40 + 28 \beta_{1} - 12 \beta_{2} + 68 \beta_{3} - 52 \beta_{4} + 20 \beta_{5} + 128 \beta_{6} + 52 \beta_{7} - 56 \beta_{8} - 32 \beta_{9} - 12 \beta_{10} - 36 \beta_{11} ) q^{76} + ( -93 - 40 \beta_{1} - 222 \beta_{2} - 93 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} - 89 \beta_{6} + 10 \beta_{7} - 98 \beta_{8} - 11 \beta_{10} - 11 \beta_{11} ) q^{77} + ( 6 + 20 \beta_{1} + 52 \beta_{2} + 200 \beta_{3} + 20 \beta_{5} + 26 \beta_{6} + 52 \beta_{7} - 8 \beta_{9} - 12 \beta_{10} - 16 \beta_{11} ) q^{78} + ( 139 + 18 \beta_{1} - 90 \beta_{2} - 31 \beta_{4} + 17 \beta_{5} - 286 \beta_{6} + 334 \beta_{7} - 203 \beta_{8} + 17 \beta_{9} + 18 \beta_{10} ) q^{79} + ( 16 - 16 \beta_{1} - 16 \beta_{4} + 80 \beta_{6} + 16 \beta_{7} - 16 \beta_{9} + 16 \beta_{11} ) q^{80} + ( 75 + 6 \beta_{1} + 471 \beta_{2} - 236 \beta_{3} + 21 \beta_{4} - 27 \beta_{5} + 96 \beta_{6} - 417 \beta_{7} + 236 \beta_{8} + 21 \beta_{9} + 21 \beta_{10} + 21 \beta_{11} ) q^{81} + ( -126 - 28 \beta_{1} - 164 \beta_{2} + 98 \beta_{3} + 42 \beta_{4} + 6 \beta_{5} - 122 \beta_{6} - 270 \beta_{7} - 234 \beta_{8} + 42 \beta_{9} + 14 \beta_{11} ) q^{82} + ( 17 + 15 \beta_{1} + 405 \beta_{2} + 35 \beta_{3} + 13 \beta_{4} - 17 \beta_{5} - 12 \beta_{6} - 162 \beta_{7} - 390 \beta_{8} - 15 \beta_{9} + 30 \beta_{10} - 17 \beta_{11} ) q^{83} + ( 28 + 20 \beta_{2} - 40 \beta_{3} - 20 \beta_{5} + 104 \beta_{6} + 28 \beta_{7} - 64 \beta_{8} - 8 \beta_{9} + 20 \beta_{11} ) q^{84} + ( -479 - 50 \beta_{1} - 69 \beta_{2} - 95 \beta_{3} - 44 \beta_{4} + 94 \beta_{6} - 384 \beta_{7} + 385 \beta_{8} + 94 \beta_{9} + 50 \beta_{11} ) q^{85} + ( 122 - 8 \beta_{1} + 702 \beta_{2} + 86 \beta_{3} - 22 \beta_{4} + 30 \beta_{6} + 36 \beta_{7} - 152 \beta_{8} + 30 \beta_{9} + 14 \beta_{10} + 8 \beta_{11} ) q^{86} + ( 499 + 37 \beta_{2} - 23 \beta_{3} + 14 \beta_{4} + 42 \beta_{5} + 132 \beta_{6} + 513 \beta_{7} - 109 \beta_{8} - 34 \beta_{9} - 14 \beta_{10} - 28 \beta_{11} ) q^{87} + ( 16 - 128 \beta_{2} - 104 \beta_{3} + 24 \beta_{4} - 16 \beta_{5} - 160 \beta_{6} - 8 \beta_{7} + 128 \beta_{8} + 40 \beta_{10} - 16 \beta_{11} ) q^{88} + ( -472 - 91 \beta_{1} + 216 \beta_{2} + 606 \beta_{3} + 24 \beta_{4} + 128 \beta_{5} + 240 \beta_{6} + 47 \beta_{7} - 57 \beta_{8} + 24 \beta_{9} - 67 \beta_{11} ) q^{89} + ( -334 - 58 \beta_{1} - 564 \beta_{2} - 402 \beta_{3} + 54 \beta_{4} + 2 \beta_{5} - 280 \beta_{6} + 174 \beta_{7} + 402 \beta_{8} + 56 \beta_{9} + 56 \beta_{10} + 54 \beta_{11} ) q^{90} + ( 152 - 32 \beta_{1} + 84 \beta_{3} - 32 \beta_{4} - 41 \beta_{5} - 225 \beta_{6} + 66 \beta_{7} + 36 \beta_{8} - 23 \beta_{9} + 41 \beta_{10} + 23 \beta_{11} ) q^{91} + ( -52 - 52 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} + 36 \beta_{5} + 72 \beta_{6} - 40 \beta_{7} - 184 \beta_{8} + 36 \beta_{9} - 52 \beta_{10} ) q^{92} + ( -219 + 53 \beta_{1} - 551 \beta_{2} + 209 \beta_{3} + 53 \beta_{5} - 385 \beta_{6} - 551 \beta_{7} - 65 \beta_{9} + 12 \beta_{10} - 3 \beta_{11} ) q^{93} + ( 288 - 62 \beta_{1} - 72 \beta_{2} - 252 \beta_{3} + 58 \beta_{4} - 58 \beta_{5} + 242 \beta_{6} - 58 \beta_{7} + 368 \beta_{8} - 10 \beta_{10} - 10 \beta_{11} ) q^{94} + ( 57 + 95 \beta_{1} - 247 \beta_{2} + 665 \beta_{3} + 38 \beta_{4} - 456 \beta_{6} + 209 \beta_{7} + 95 \beta_{8} - 19 \beta_{9} - 57 \beta_{10} + 19 \beta_{11} ) q^{95} + ( -32 - 32 \beta_{1} - 64 \beta_{2} - 32 \beta_{3} ) q^{96} + ( -473 + 4 \beta_{1} + 173 \beta_{2} - 306 \beta_{3} + 4 \beta_{5} + 642 \beta_{6} + 173 \beta_{7} - 72 \beta_{9} + 68 \beta_{10} - 125 \beta_{11} ) q^{97} + ( 112 + 20 \beta_{1} - 78 \beta_{2} - 14 \beta_{4} - 26 \beta_{5} - 140 \beta_{6} + 128 \beta_{7} - 462 \beta_{8} - 26 \beta_{9} + 20 \beta_{10} ) q^{98} + ( 443 + 53 \beta_{1} + 313 \beta_{3} + 53 \beta_{4} + 138 \beta_{5} - 52 \beta_{6} + 396 \beta_{7} + 183 \beta_{8} + 55 \beta_{9} - 138 \beta_{10} - 55 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 9q^{3} - 18q^{6} + 21q^{7} - 48q^{8} - 27q^{9} + O(q^{10})$$ $$12q - 9q^{3} - 18q^{6} + 21q^{7} - 48q^{8} - 27q^{9} - 9q^{11} + 36q^{12} + 39q^{13} - 138q^{14} + 423q^{15} + 69q^{17} + 132q^{18} - 462q^{19} - 216q^{20} - 279q^{21} + 204q^{22} - 66q^{23} - 72q^{24} + 342q^{25} + 48q^{26} + 189q^{27} + 192q^{28} + 159q^{29} + 72q^{31} - 1560q^{33} + 408q^{34} - 135q^{35} - 108q^{36} + 1116q^{37} - 294q^{38} - 1248q^{39} + 147q^{41} + 414q^{42} - 117q^{43} + 408q^{44} + 1296q^{45} + 528q^{46} + 783q^{47} + 288q^{48} + 1413q^{49} - 354q^{50} - 2301q^{51} - 348q^{52} - 249q^{53} - 540q^{54} + 2187q^{55} - 336q^{56} - 2670q^{57} - 1932q^{58} - 4248q^{59} + 324q^{60} + 3114q^{61} - 438q^{62} + 363q^{63} - 384q^{64} + 495q^{65} + 822q^{66} + 3060q^{67} + 408q^{68} - 237q^{69} - 270q^{70} + 1686q^{71} + 432q^{72} + 1626q^{73} + 90q^{74} - 1854q^{75} - 1416q^{77} - 108q^{78} - 327q^{79} + 3483q^{81} + 294q^{82} + 927q^{83} + 204q^{84} - 3294q^{85} + 1188q^{86} + 2892q^{87} - 72q^{88} - 6366q^{89} - 5076q^{90} + 840q^{91} - 156q^{92} + 870q^{93} + 3432q^{94} + 513q^{95} - 576q^{96} - 8052q^{97} + 378q^{98} + 4494q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} - 135 x^{10} + 730 x^{9} + 7953 x^{8} - 36258 x^{7} - 262940 x^{6} + 918855 x^{5} + 5157591 x^{4} - 11890401 x^{3} - 56759508 x^{2} + 62864118 x + 272110107$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-177221217078 \nu^{11} + 3257235752468 \nu^{10} + 7448859189417 \nu^{9} - 348800270203041 \nu^{8} + 246725060413890 \nu^{7} + 15490888685337464 \nu^{6} - 19811196354376926 \nu^{5} - 357754300688799514 \nu^{4} + 400473915693733842 \nu^{3} + 4306750761297997378 \nu^{2} - 2692306104561693094 \nu - 21338093845933036168$$$$)/ 207201391860211327$$ $$\beta_{3}$$ $$=$$ $$($$$$177221217078 \nu^{11} + 1307802364610 \nu^{10} - 30274049774807 \nu^{9} - 164426429455098 \nu^{8} + 1943132881731006 \nu^{7} + 8679404020188818 \nu^{6} - 60460050360167694 \nu^{5} - 238663501216099822 \nu^{4} + 928747583953979022 \nu^{3} + 3391717877058441844 \nu^{2} - 5684523648618559763 \nu - 19685341428196901362$$$$)/ 207201391860211327$$ $$\beta_{4}$$ $$=$$ $$($$$$-488144071095 \nu^{11} + 17846740004059 \nu^{10} - 13839005362779 \nu^{9} - 2106984070175911 \nu^{8} + 4541779588170822 \nu^{7} + 102961725378166441 \nu^{6} - 217798892218728099 \nu^{5} - 2625515013443047084 \nu^{4} + 4208642241669180665 \nu^{3} + 35298117216643037198 \nu^{2} - 30071865827575778859 \nu - 200899777020391137346$$$$)/ 207201391860211327$$ $$\beta_{5}$$ $$=$$ $$($$$$488144071095 \nu^{11} + 12477155222014 \nu^{10} - 137780470767586 \nu^{9} - 1508975589988942 \nu^{8} + 10831775909270780 \nu^{7} + 78118444919969751 \nu^{6} - 379885864716473617 \nu^{5} - 2156767180008845078 \nu^{4} + 6370302460492549118 \nu^{3} + 31574050688047386454 \nu^{2} - 42191996299443789347 \nu - 194202797254829741988$$$$)/ 207201391860211327$$ $$\beta_{6}$$ $$=$$ $$($$$$-545356750193 \nu^{11} + 639178329540 \nu^{10} + 77274249510048 \nu^{9} - 43408610678418 \nu^{8} - 4452252157863755 \nu^{7} - 199135495484657 \nu^{6} + 130514673620858285 \nu^{5} + 80929955889112716 \nu^{4} - 1946446810652341253 \nu^{3} - 2244991753408961063 \nu^{2} + 11804378059999411394 \nu + 19269854158760561501$$$$)/ 207201391860211327$$ $$\beta_{7}$$ $$=$$ $$($$$$1581666 \nu^{11} - 8699163 \nu^{10} - 191254825 \nu^{9} + 925890435 \nu^{8} + 9722082222 \nu^{7} - 38409003948 \nu^{6} - 260049871314 \nu^{5} + 748351094985 \nu^{4} + 3667777839865 \nu^{3} - 6269852665791 \nu^{2} - 21765274073007 \nu + 11710200330886$$$$)/ 486606417103$$ $$\beta_{8}$$ $$=$$ $$($$$$722577967271 \nu^{11} - 4051943557973 \nu^{10} - 83945461319640 \nu^{9} + 426412366504371 \nu^{8} + 4064047270679267 \nu^{7} - 17527628089879562 \nu^{6} - 103497407053708736 \nu^{5} + 342676177232219537 \nu^{4} + 1402092497979595547 \nu^{3} - 2996806964224757833 \nu^{2} - 8103858975645019699 \nu + 7404279427818802783$$$$)/ 207201391860211327$$ $$\beta_{9}$$ $$=$$ $$($$$$882100559090 \nu^{11} + 8843624294225 \nu^{10} - 174667730262753 \nu^{9} - 1173088304769666 \nu^{8} + 12516666641744675 \nu^{7} + 66554394063895991 \nu^{6} - 430697507264496290 \nu^{5} - 1992730817819811185 \nu^{4} + 7284724960989936728 \nu^{3} + 31203967995735408752 \nu^{2} - 48949073051985878985 \nu - 202532641899861005459$$$$)/ 207201391860211327$$ $$\beta_{10}$$ $$=$$ $$($$$$-1648551699807 \nu^{11} + 11116259193530 \nu^{10} + 165757370191268 \nu^{9} - 1065326603875755 \nu^{8} - 6944287407575079 \nu^{7} + 40202711062797866 \nu^{6} + 151876342577052782 \nu^{5} - 733168791455953581 \nu^{4} - 1726174510506912575 \nu^{3} + 6362770351786971041 \nu^{2} + 7777903016411569028 \nu - 21134063485749103201$$$$)/ 207201391860211327$$ $$\beta_{11}$$ $$=$$ $$($$$$-1648551699807 \nu^{11} + 7017809504347 \nu^{10} + 186249618637183 \nu^{9} - 654710361086352 \nu^{8} - 8709705869408181 \nu^{7} + 22740043057351849 \nu^{6} + 210529378653279533 \nu^{5} - 334306486848652462 \nu^{4} - 2623754819217589847 \nu^{3} + 1262244190577485046 \nu^{2} + 13337295222073937409 \nu + 9268488754807344483$$$$)/ 207201391860211327$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{11} + \beta_{10} - 3 \beta_{8} - 3 \beta_{6} - \beta_{3} - 4 \beta_{2} + \beta_{1} + 26$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{10} - 15 \beta_{8} + 10 \beta_{7} + 6 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 28 \beta_{3} - 25 \beta_{2} + 25 \beta_{1} + 32$$ $$\nu^{4}$$ $$=$$ $$-53 \beta_{11} + 59 \beta_{10} + 10 \beta_{9} - 217 \beta_{8} + 15 \beta_{7} - 165 \beta_{6} - 3 \beta_{5} - 15 \beta_{4} + 37 \beta_{3} - 256 \beta_{2} + 44 \beta_{1} + 698$$ $$\nu^{5}$$ $$=$$ $$-81 \beta_{11} + 194 \beta_{10} + 25 \beta_{9} - 1321 \beta_{8} + 892 \beta_{7} + 381 \beta_{6} + 162 \beta_{5} - 207 \beta_{4} + 1665 \beta_{3} - 1414 \beta_{2} + 615 \beta_{1} + 1854$$ $$\nu^{6}$$ $$=$$ $$-2280 \beta_{11} + 2604 \beta_{10} + 917 \beta_{9} - 11518 \beta_{8} + 2205 \beta_{7} - 5675 \beta_{6} - 219 \beta_{5} - 1296 \beta_{4} + 5259 \beta_{3} - 11343 \beta_{2} + 1302 \beta_{1} + 19896$$ $$\nu^{7}$$ $$=$$ $$-6561 \beta_{11} + 9574 \beta_{10} + 3122 \beta_{9} - 76494 \beta_{8} + 52916 \beta_{7} + 19598 \beta_{6} + 5456 \beta_{5} - 10601 \beta_{4} + 78989 \beta_{3} - 60371 \beta_{2} + 14109 \beta_{1} + 81501$$ $$\nu^{8}$$ $$=$$ $$-93098 \beta_{11} + 103652 \beta_{10} + 56038 \beta_{9} - 531544 \beta_{8} + 177506 \beta_{7} - 126516 \beta_{6} - 14142 \beta_{5} - 73372 \beta_{4} + 339489 \beta_{3} - 419971 \beta_{2} + 26559 \beta_{1} + 609442$$ $$\nu^{9}$$ $$=$$ $$-366048 \beta_{11} + 423666 \beta_{10} + 233544 \beta_{9} - 3638850 \beta_{8} + 2627538 \beta_{7} + 1015026 \beta_{6} + 112374 \beta_{5} - 475506 \beta_{4} + 3478249 \beta_{3} - 2250730 \beta_{2} + 261745 \beta_{1} + 3220296$$ $$\nu^{10}$$ $$=$$ $$-3739994 \beta_{11} + 3951167 \beta_{10} + 2861082 \beta_{9} - 22427599 \beta_{8} + 10768116 \beta_{7} - 48010 \beta_{6} - 902652 \beta_{5} - 3405306 \beta_{4} + 17512846 \beta_{3} - 13691236 \beta_{2} + 64789 \beta_{1} + 19795826$$ $$\nu^{11}$$ $$=$$ $$-17577635 \beta_{11} + 17659182 \beta_{10} + 13629198 \beta_{9} - 153343365 \beta_{8} + 118000897 \beta_{7} + 53163670 \beta_{6} - 854642 \beta_{5} - 19566517 \beta_{4} + 147712096 \beta_{3} - 75222735 \beta_{2} + 1401338 \beta_{1} + 119730930$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$\chi(n)$$ $$-\beta_{6}$$

## Embeddings

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

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

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 5.05412 − 0.342020i −4.05412 − 0.342020i 5.30460 − 0.984808i −4.30460 − 0.984808i 5.30460 + 0.984808i −4.30460 + 0.984808i 5.05412 + 0.342020i −4.05412 + 0.342020i −5.28151 + 0.642788i 6.28151 + 0.642788i −5.28151 − 0.642788i 6.28151 − 0.642788i
−1.87939 + 0.684040i −1.54081 + 8.73839i 3.06418 2.57115i −15.3487 12.8791i −3.08163 17.4768i 4.71270 + 8.16264i −4.00000 + 6.92820i −48.6136 17.6939i 37.6558 + 13.7056i
5.2 −1.87939 + 0.684040i 0.0408138 0.231467i 3.06418 2.57115i 8.45426 + 7.09396i 0.0816277 + 0.462933i 9.26682 + 16.0506i −4.00000 + 6.92820i 25.3198 + 9.21565i −20.7414 7.54924i
9.1 0.347296 + 1.96962i −4.43054 + 3.71766i −3.75877 + 1.36808i −5.82452 2.11995i −8.86108 7.43533i −5.61124 + 9.71895i −4.00000 6.92820i 1.12015 6.35271i 2.15265 12.2083i
9.2 0.347296 + 1.96962i 2.93054 2.45902i −3.75877 + 1.36808i 14.2818 + 5.19813i 5.86108 + 4.91803i −0.806634 + 1.39713i −4.00000 6.92820i −2.14719 + 12.1773i −5.27832 + 29.9349i
17.1 0.347296 1.96962i −4.43054 3.71766i −3.75877 1.36808i −5.82452 + 2.11995i −8.86108 + 7.43533i −5.61124 9.71895i −4.00000 + 6.92820i 1.12015 + 6.35271i 2.15265 + 12.2083i
17.2 0.347296 1.96962i 2.93054 + 2.45902i −3.75877 1.36808i 14.2818 5.19813i 5.86108 4.91803i −0.806634 1.39713i −4.00000 + 6.92820i −2.14719 12.1773i −5.27832 29.9349i
23.1 −1.87939 0.684040i −1.54081 8.73839i 3.06418 + 2.57115i −15.3487 + 12.8791i −3.08163 + 17.4768i 4.71270 8.16264i −4.00000 6.92820i −48.6136 + 17.6939i 37.6558 13.7056i
23.2 −1.87939 0.684040i 0.0408138 + 0.231467i 3.06418 + 2.57115i 8.45426 7.09396i 0.0816277 0.462933i 9.26682 16.0506i −4.00000 6.92820i 25.3198 9.21565i −20.7414 + 7.54924i
25.1 1.53209 1.28558i −6.18284 2.25037i 0.694593 3.93923i −1.97089 11.1775i −12.3657 + 4.50074i 4.35993 7.55162i −4.00000 6.92820i 12.4801 + 10.4721i −17.3890 14.5911i
25.2 1.53209 1.28558i 4.68284 + 1.70441i 0.694593 3.93923i 0.408053 + 2.31419i 9.36568 3.40883i −1.42158 + 2.46225i −4.00000 6.92820i −1.65925 1.39227i 3.60023 + 3.02095i
35.1 1.53209 + 1.28558i −6.18284 + 2.25037i 0.694593 + 3.93923i −1.97089 + 11.1775i −12.3657 4.50074i 4.35993 + 7.55162i −4.00000 + 6.92820i 12.4801 10.4721i −17.3890 + 14.5911i
35.2 1.53209 + 1.28558i 4.68284 1.70441i 0.694593 + 3.93923i 0.408053 2.31419i 9.36568 + 3.40883i −1.42158 2.46225i −4.00000 + 6.92820i −1.65925 + 1.39227i 3.60023 3.02095i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 35.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.4.e.a 12
19.e even 9 1 inner 38.4.e.a 12
19.e even 9 1 722.4.a.p 6
19.f odd 18 1 722.4.a.o 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.e.a 12 1.a even 1 1 trivial
38.4.e.a 12 19.e even 9 1 inner
722.4.a.o 6 19.f odd 18 1
722.4.a.p 6 19.e even 9 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 8 T^{3} + 64 T^{6} )^{2}$$
$3$ $$1 + 9 T + 54 T^{2} + 261 T^{3} + 567 T^{4} - 396 T^{5} - 4724 T^{6} + 5823 T^{7} + 507654 T^{8} + 2406897 T^{9} - 1345086 T^{10} - 77064138 T^{11} - 485342900 T^{12} - 2080731726 T^{13} - 980567694 T^{14} + 47374953651 T^{15} + 269788149414 T^{16} + 83553685461 T^{17} - 1830174390036 T^{18} - 4142299868388 T^{19} + 160137547184727 T^{20} + 1990280943581607 T^{21} + 11118121133111046 T^{22} + 50031545098999707 T^{23} + 150094635296999121 T^{24}$$
$5$ $$1 - 171 T^{2} - 432 T^{3} + 684 T^{4} + 189297 T^{5} + 2035721 T^{6} - 20411757 T^{7} - 39405582 T^{8} - 2528638560 T^{9} - 4712335920 T^{10} + 292985586000 T^{11} - 629941358634 T^{12} + 36623198250000 T^{13} - 73630248750000 T^{14} - 4938747187500000 T^{15} - 9620503417968750 T^{16} - 622917388916015625 T^{17} + 7765659332275390625 T^{18} + 90263843536376953125 T^{19} + 40769577026367187500 T^{20} -$$$$32\!\cdots\!00$$$$T^{21} -$$$$15\!\cdots\!75$$$$T^{22} +$$$$14\!\cdots\!25$$$$T^{24}$$
$7$ $$1 - 21 T - 1515 T^{2} + 24780 T^{3} + 1512579 T^{4} - 17880063 T^{5} - 1075565731 T^{6} + 8262030114 T^{7} + 606007378278 T^{8} - 2582097624714 T^{9} - 275006022177042 T^{10} + 351692577138024 T^{11} + 103612362769606380 T^{12} + 120630553958342232 T^{13} - 32354183503106814258 T^{14} -$$$$10\!\cdots\!98$$$$T^{15} +$$$$83\!\cdots\!78$$$$T^{16} +$$$$39\!\cdots\!02$$$$T^{17} -$$$$17\!\cdots\!19$$$$T^{18} -$$$$99\!\cdots\!41$$$$T^{19} +$$$$28\!\cdots\!79$$$$T^{20} +$$$$16\!\cdots\!40$$$$T^{21} -$$$$34\!\cdots\!35$$$$T^{22} -$$$$16\!\cdots\!47$$$$T^{23} +$$$$26\!\cdots\!01$$$$T^{24}$$
$11$ $$1 + 9 T - 3519 T^{2} + 24496 T^{3} + 5114331 T^{4} - 110124705 T^{5} - 2271860931 T^{6} + 52499450874 T^{7} + 731386484418 T^{8} + 207758908815234 T^{9} - 13017969765518466 T^{10} - 204189988973228292 T^{11} + 30443947856053596940 T^{12} -$$$$27\!\cdots\!52$$$$T^{13} -$$$$23\!\cdots\!26$$$$T^{14} +$$$$48\!\cdots\!94$$$$T^{15} +$$$$22\!\cdots\!78$$$$T^{16} +$$$$21\!\cdots\!74$$$$T^{17} -$$$$12\!\cdots\!11$$$$T^{18} -$$$$81\!\cdots\!55$$$$T^{19} +$$$$50\!\cdots\!71$$$$T^{20} +$$$$32\!\cdots\!16$$$$T^{21} -$$$$61\!\cdots\!19$$$$T^{22} +$$$$20\!\cdots\!79$$$$T^{23} +$$$$30\!\cdots\!61$$$$T^{24}$$
$13$ $$1 - 39 T + 3093 T^{2} - 98440 T^{3} + 11349087 T^{4} - 378701979 T^{5} + 22862720507 T^{6} - 1116317647566 T^{7} + 56385479661678 T^{8} - 3687796202409564 T^{9} + 159033549430554174 T^{10} - 10099287187718310156 T^{11} +$$$$30\!\cdots\!88$$$$T^{12} -$$$$22\!\cdots\!32$$$$T^{13} +$$$$76\!\cdots\!66$$$$T^{14} -$$$$39\!\cdots\!72$$$$T^{15} +$$$$13\!\cdots\!18$$$$T^{16} -$$$$57\!\cdots\!62$$$$T^{17} +$$$$25\!\cdots\!03$$$$T^{18} -$$$$93\!\cdots\!27$$$$T^{19} +$$$$61\!\cdots\!07$$$$T^{20} -$$$$11\!\cdots\!80$$$$T^{21} +$$$$81\!\cdots\!57$$$$T^{22} -$$$$22\!\cdots\!67$$$$T^{23} +$$$$12\!\cdots\!41$$$$T^{24}$$
$17$ $$1 - 69 T + 6573 T^{2} + 298204 T^{3} - 55154319 T^{4} + 4598370171 T^{5} - 148240032933 T^{6} - 16787667391872 T^{7} + 1892272804048104 T^{8} - 125048706972526476 T^{9} + 913900858267910370 T^{10} +$$$$57\!\cdots\!02$$$$T^{11} -$$$$51\!\cdots\!92$$$$T^{12} +$$$$28\!\cdots\!26$$$$T^{13} +$$$$22\!\cdots\!30$$$$T^{14} -$$$$14\!\cdots\!72$$$$T^{15} +$$$$11\!\cdots\!44$$$$T^{16} -$$$$48\!\cdots\!96$$$$T^{17} -$$$$20\!\cdots\!97$$$$T^{18} +$$$$31\!\cdots\!07$$$$T^{19} -$$$$18\!\cdots\!99$$$$T^{20} +$$$$49\!\cdots\!92$$$$T^{21} +$$$$53\!\cdots\!77$$$$T^{22} -$$$$27\!\cdots\!53$$$$T^{23} +$$$$19\!\cdots\!81$$$$T^{24}$$
$19$ $$1 + 462 T + 119769 T^{2} + 21623273 T^{3} + 2994678027 T^{4} + 332359765731 T^{5} + 30289255560054 T^{6} + 2279655633148929 T^{7} + 140887266091556787 T^{8} + 6977564182816810667 T^{9} +$$$$26\!\cdots\!09$$$$T^{10} +$$$$70\!\cdots\!38$$$$T^{11} +$$$$10\!\cdots\!41$$$$T^{12}$$
$23$ $$1 + 66 T - 9393 T^{2} + 671695 T^{3} + 134377509 T^{4} - 14874284241 T^{5} + 71997450171 T^{6} + 263342907791031 T^{7} - 15774109169407374 T^{8} - 4876569355971214692 T^{9} +$$$$29\!\cdots\!14$$$$T^{10} +$$$$83\!\cdots\!47$$$$T^{11} -$$$$84\!\cdots\!52$$$$T^{12} +$$$$10\!\cdots\!49$$$$T^{13} +$$$$42\!\cdots\!46$$$$T^{14} -$$$$87\!\cdots\!96$$$$T^{15} -$$$$34\!\cdots\!54$$$$T^{16} +$$$$70\!\cdots\!17$$$$T^{17} +$$$$23\!\cdots\!99$$$$T^{18} -$$$$58\!\cdots\!43$$$$T^{19} +$$$$64\!\cdots\!69$$$$T^{20} +$$$$39\!\cdots\!65$$$$T^{21} -$$$$66\!\cdots\!57$$$$T^{22} +$$$$57\!\cdots\!78$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$
$29$ $$1 - 159 T + 36549 T^{2} - 3331656 T^{3} + 421841088 T^{4} - 94908073713 T^{5} + 5535241134320 T^{6} + 625908025808775 T^{7} + 30175667319970098 T^{8} - 48469236719521320171 T^{9} +$$$$14\!\cdots\!33$$$$T^{10} -$$$$23\!\cdots\!52$$$$T^{11} +$$$$17\!\cdots\!06$$$$T^{12} -$$$$58\!\cdots\!28$$$$T^{13} +$$$$86\!\cdots\!93$$$$T^{14} -$$$$70\!\cdots\!99$$$$T^{15} +$$$$10\!\cdots\!18$$$$T^{16} +$$$$54\!\cdots\!75$$$$T^{17} +$$$$11\!\cdots\!20$$$$T^{18} -$$$$48\!\cdots\!77$$$$T^{19} +$$$$52\!\cdots\!28$$$$T^{20} -$$$$10\!\cdots\!04$$$$T^{21} +$$$$27\!\cdots\!49$$$$T^{22} -$$$$28\!\cdots\!51$$$$T^{23} +$$$$44\!\cdots\!21$$$$T^{24}$$
$31$ $$1 - 72 T - 84360 T^{2} - 4099822 T^{3} + 4645966785 T^{4} + 582123802836 T^{5} - 97979947426741 T^{6} - 42244835246634435 T^{7} - 818163153401404386 T^{8} +$$$$13\!\cdots\!80$$$$T^{9} +$$$$20\!\cdots\!33$$$$T^{10} -$$$$20\!\cdots\!07$$$$T^{11} -$$$$78\!\cdots\!40$$$$T^{12} -$$$$60\!\cdots\!37$$$$T^{13} +$$$$18\!\cdots\!73$$$$T^{14} +$$$$35\!\cdots\!80$$$$T^{15} -$$$$64\!\cdots\!46$$$$T^{16} -$$$$99\!\cdots\!85$$$$T^{17} -$$$$68\!\cdots\!81$$$$T^{18} +$$$$12\!\cdots\!16$$$$T^{19} +$$$$28\!\cdots\!85$$$$T^{20} -$$$$75\!\cdots\!42$$$$T^{21} -$$$$46\!\cdots\!60$$$$T^{22} -$$$$11\!\cdots\!52$$$$T^{23} +$$$$48\!\cdots\!81$$$$T^{24}$$
$37$ $$( 1 - 558 T + 298089 T^{2} - 113545809 T^{3} + 38178310329 T^{4} - 10429273769733 T^{5} + 2577693076919434 T^{6} - 528274004258285649 T^{7} + 97955099062112778561 T^{8} -$$$$14\!\cdots\!93$$$$T^{9} +$$$$19\!\cdots\!09$$$$T^{10} -$$$$18\!\cdots\!94$$$$T^{11} +$$$$16\!\cdots\!29$$$$T^{12} )^{2}$$
$41$ $$1 - 147 T + 44007 T^{2} + 21769246 T^{3} - 2350025733 T^{4} + 234418847265 T^{5} + 457706944154289 T^{6} - 9303997209190392 T^{7} - 6140319093154185000 T^{8} +$$$$10\!\cdots\!34$$$$T^{9} -$$$$12\!\cdots\!74$$$$T^{10} -$$$$45\!\cdots\!46$$$$T^{11} +$$$$16\!\cdots\!04$$$$T^{12} -$$$$31\!\cdots\!66$$$$T^{13} -$$$$59\!\cdots\!34$$$$T^{14} +$$$$34\!\cdots\!74$$$$T^{15} -$$$$13\!\cdots\!00$$$$T^{16} -$$$$14\!\cdots\!92$$$$T^{17} +$$$$49\!\cdots\!69$$$$T^{18} +$$$$17\!\cdots\!65$$$$T^{19} -$$$$11\!\cdots\!13$$$$T^{20} +$$$$76\!\cdots\!26$$$$T^{21} +$$$$10\!\cdots\!07$$$$T^{22} -$$$$24\!\cdots\!87$$$$T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$43$ $$1 + 117 T - 100119 T^{2} + 38915762 T^{3} + 2994100635 T^{4} - 3671912050095 T^{5} + 1310121343946135 T^{6} - 70375225814813832 T^{7} - 74777917746500954580 T^{8} +$$$$41\!\cdots\!76$$$$T^{9} -$$$$21\!\cdots\!80$$$$T^{10} -$$$$11\!\cdots\!68$$$$T^{11} +$$$$88\!\cdots\!32$$$$T^{12} -$$$$89\!\cdots\!76$$$$T^{13} -$$$$13\!\cdots\!20$$$$T^{14} +$$$$20\!\cdots\!68$$$$T^{15} -$$$$29\!\cdots\!80$$$$T^{16} -$$$$22\!\cdots\!24$$$$T^{17} +$$$$33\!\cdots\!15$$$$T^{18} -$$$$73\!\cdots\!85$$$$T^{19} +$$$$47\!\cdots\!35$$$$T^{20} +$$$$49\!\cdots\!34$$$$T^{21} -$$$$10\!\cdots\!31$$$$T^{22} +$$$$93\!\cdots\!31$$$$T^{23} +$$$$63\!\cdots\!01$$$$T^{24}$$
$47$ $$1 - 783 T + 538857 T^{2} - 254728868 T^{3} + 112205777895 T^{4} - 37288296975249 T^{5} + 10875425952712257 T^{6} - 1895616375918277806 T^{7} + 46254528927632446254 T^{8} +$$$$26\!\cdots\!66$$$$T^{9} -$$$$15\!\cdots\!14$$$$T^{10} +$$$$68\!\cdots\!52$$$$T^{11} -$$$$22\!\cdots\!72$$$$T^{12} +$$$$70\!\cdots\!96$$$$T^{13} -$$$$16\!\cdots\!06$$$$T^{14} +$$$$29\!\cdots\!22$$$$T^{15} +$$$$53\!\cdots\!14$$$$T^{16} -$$$$22\!\cdots\!58$$$$T^{17} +$$$$13\!\cdots\!73$$$$T^{18} -$$$$48\!\cdots\!03$$$$T^{19} +$$$$15\!\cdots\!95$$$$T^{20} -$$$$35\!\cdots\!84$$$$T^{21} +$$$$78\!\cdots\!93$$$$T^{22} -$$$$11\!\cdots\!41$$$$T^{23} +$$$$15\!\cdots\!21$$$$T^{24}$$
$53$ $$1 + 249 T + 105357 T^{2} + 53055920 T^{3} + 56278965819 T^{4} + 10000173124845 T^{5} + 3156327827474091 T^{6} + 2244063380209899702 T^{7} +$$$$94\!\cdots\!94$$$$T^{8} +$$$$24\!\cdots\!32$$$$T^{9} +$$$$31\!\cdots\!86$$$$T^{10} +$$$$47\!\cdots\!40$$$$T^{11} +$$$$98\!\cdots\!48$$$$T^{12} +$$$$70\!\cdots\!80$$$$T^{13} +$$$$69\!\cdots\!94$$$$T^{14} +$$$$79\!\cdots\!56$$$$T^{15} +$$$$46\!\cdots\!54$$$$T^{16} +$$$$16\!\cdots\!14$$$$T^{17} +$$$$34\!\cdots\!99$$$$T^{18} +$$$$16\!\cdots\!85$$$$T^{19} +$$$$13\!\cdots\!39$$$$T^{20} +$$$$19\!\cdots\!40$$$$T^{21} +$$$$56\!\cdots\!93$$$$T^{22} +$$$$19\!\cdots\!77$$$$T^{23} +$$$$11\!\cdots\!21$$$$T^{24}$$
$59$ $$1 + 4248 T + 8725923 T^{2} + 11515262701 T^{3} + 10887706714419 T^{4} + 7727213299360905 T^{5} + 4135284193453945767 T^{6} +$$$$15\!\cdots\!07$$$$T^{7} +$$$$29\!\cdots\!98$$$$T^{8} -$$$$11\!\cdots\!96$$$$T^{9} -$$$$15\!\cdots\!02$$$$T^{10} -$$$$10\!\cdots\!07$$$$T^{11} -$$$$50\!\cdots\!20$$$$T^{12} -$$$$20\!\cdots\!53$$$$T^{13} -$$$$66\!\cdots\!82$$$$T^{14} -$$$$10\!\cdots\!44$$$$T^{15} +$$$$52\!\cdots\!38$$$$T^{16} +$$$$57\!\cdots\!93$$$$T^{17} +$$$$31\!\cdots\!07$$$$T^{18} +$$$$11\!\cdots\!95$$$$T^{19} +$$$$34\!\cdots\!59$$$$T^{20} +$$$$74\!\cdots\!19$$$$T^{21} +$$$$11\!\cdots\!23$$$$T^{22} +$$$$11\!\cdots\!92$$$$T^{23} +$$$$56\!\cdots\!41$$$$T^{24}$$
$61$ $$1 - 3114 T + 4298859 T^{2} - 3204919049 T^{3} + 998006352417 T^{4} + 482660655435063 T^{5} - 719435635956944909 T^{6} +$$$$34\!\cdots\!33$$$$T^{7} -$$$$35\!\cdots\!32$$$$T^{8} -$$$$50\!\cdots\!02$$$$T^{9} +$$$$27\!\cdots\!08$$$$T^{10} -$$$$17\!\cdots\!79$$$$T^{11} -$$$$25\!\cdots\!48$$$$T^{12} -$$$$39\!\cdots\!99$$$$T^{13} +$$$$13\!\cdots\!88$$$$T^{14} -$$$$59\!\cdots\!82$$$$T^{15} -$$$$94\!\cdots\!72$$$$T^{16} +$$$$21\!\cdots\!33$$$$T^{17} -$$$$98\!\cdots\!29$$$$T^{18} +$$$$14\!\cdots\!43$$$$T^{19} +$$$$70\!\cdots\!97$$$$T^{20} -$$$$51\!\cdots\!29$$$$T^{21} +$$$$15\!\cdots\!59$$$$T^{22} -$$$$25\!\cdots\!34$$$$T^{23} +$$$$18\!\cdots\!61$$$$T^{24}$$
$67$ $$1 - 3060 T + 4343541 T^{2} - 3539177707 T^{3} + 1674046544745 T^{4} - 364838652101685 T^{5} - 33128397443652253 T^{6} + 59458849334550877089 T^{7} -$$$$73\!\cdots\!64$$$$T^{8} +$$$$81\!\cdots\!52$$$$T^{9} -$$$$50\!\cdots\!24$$$$T^{10} +$$$$16\!\cdots\!97$$$$T^{11} -$$$$43\!\cdots\!88$$$$T^{12} +$$$$48\!\cdots\!11$$$$T^{13} -$$$$45\!\cdots\!56$$$$T^{14} +$$$$22\!\cdots\!44$$$$T^{15} -$$$$60\!\cdots\!04$$$$T^{16} +$$$$14\!\cdots\!27$$$$T^{17} -$$$$24\!\cdots\!77$$$$T^{18} -$$$$81\!\cdots\!95$$$$T^{19} +$$$$11\!\cdots\!45$$$$T^{20} -$$$$71\!\cdots\!61$$$$T^{21} +$$$$26\!\cdots\!09$$$$T^{22} -$$$$55\!\cdots\!20$$$$T^{23} +$$$$54\!\cdots\!81$$$$T^{24}$$
$71$ $$1 - 1686 T + 1303347 T^{2} - 332046979 T^{3} - 395700826839 T^{4} + 496154621895921 T^{5} - 192091304753971689 T^{6} -$$$$10\!\cdots\!95$$$$T^{7} +$$$$20\!\cdots\!46$$$$T^{8} -$$$$13\!\cdots\!76$$$$T^{9} +$$$$27\!\cdots\!34$$$$T^{10} +$$$$26\!\cdots\!85$$$$T^{11} -$$$$28\!\cdots\!24$$$$T^{12} +$$$$96\!\cdots\!35$$$$T^{13} +$$$$34\!\cdots\!14$$$$T^{14} -$$$$59\!\cdots\!56$$$$T^{15} +$$$$32\!\cdots\!86$$$$T^{16} -$$$$63\!\cdots\!45$$$$T^{17} -$$$$40\!\cdots\!29$$$$T^{18} +$$$$37\!\cdots\!91$$$$T^{19} -$$$$10\!\cdots\!59$$$$T^{20} -$$$$32\!\cdots\!89$$$$T^{21} +$$$$44\!\cdots\!47$$$$T^{22} -$$$$20\!\cdots\!46$$$$T^{23} +$$$$44\!\cdots\!21$$$$T^{24}$$
$73$ $$1 - 1626 T + 1342599 T^{2} - 77826027 T^{3} - 482426009259 T^{4} + 337597036991529 T^{5} + 152387349969643313 T^{6} -$$$$21\!\cdots\!53$$$$T^{7} +$$$$80\!\cdots\!70$$$$T^{8} +$$$$61\!\cdots\!48$$$$T^{9} -$$$$37\!\cdots\!06$$$$T^{10} -$$$$65\!\cdots\!35$$$$T^{11} +$$$$17\!\cdots\!44$$$$T^{12} -$$$$25\!\cdots\!95$$$$T^{13} -$$$$56\!\cdots\!34$$$$T^{14} +$$$$35\!\cdots\!24$$$$T^{15} +$$$$18\!\cdots\!70$$$$T^{16} -$$$$18\!\cdots\!21$$$$T^{17} +$$$$52\!\cdots\!97$$$$T^{18} +$$$$45\!\cdots\!17$$$$T^{19} -$$$$25\!\cdots\!19$$$$T^{20} -$$$$15\!\cdots\!19$$$$T^{21} +$$$$10\!\cdots\!51$$$$T^{22} -$$$$50\!\cdots\!58$$$$T^{23} +$$$$12\!\cdots\!61$$$$T^{24}$$
$79$ $$1 + 327 T - 352623 T^{2} + 85160886 T^{3} - 74868578124 T^{4} - 561682606508301 T^{5} - 10832178836590636 T^{6} +$$$$15\!\cdots\!35$$$$T^{7} -$$$$13\!\cdots\!24$$$$T^{8} +$$$$11\!\cdots\!49$$$$T^{9} +$$$$69\!\cdots\!71$$$$T^{10} -$$$$11\!\cdots\!46$$$$T^{11} -$$$$30\!\cdots\!66$$$$T^{12} -$$$$55\!\cdots\!94$$$$T^{13} +$$$$16\!\cdots\!91$$$$T^{14} +$$$$13\!\cdots\!31$$$$T^{15} -$$$$82\!\cdots\!84$$$$T^{16} +$$$$46\!\cdots\!65$$$$T^{17} -$$$$15\!\cdots\!96$$$$T^{18} -$$$$39\!\cdots\!79$$$$T^{19} -$$$$26\!\cdots\!44$$$$T^{20} +$$$$14\!\cdots\!74$$$$T^{21} -$$$$29\!\cdots\!23$$$$T^{22} +$$$$13\!\cdots\!53$$$$T^{23} +$$$$20\!\cdots\!21$$$$T^{24}$$
$83$ $$1 - 927 T - 1910169 T^{2} + 1162039358 T^{3} + 2637336171501 T^{4} - 771225603730341 T^{5} - 2584240489804536525 T^{6} +$$$$27\!\cdots\!66$$$$T^{7} +$$$$19\!\cdots\!86$$$$T^{8} -$$$$17\!\cdots\!40$$$$T^{9} -$$$$12\!\cdots\!02$$$$T^{10} -$$$$85\!\cdots\!76$$$$T^{11} +$$$$72\!\cdots\!04$$$$T^{12} -$$$$48\!\cdots\!12$$$$T^{13} -$$$$40\!\cdots\!38$$$$T^{14} -$$$$33\!\cdots\!20$$$$T^{15} +$$$$20\!\cdots\!46$$$$T^{16} +$$$$16\!\cdots\!62$$$$T^{17} -$$$$90\!\cdots\!25$$$$T^{18} -$$$$15\!\cdots\!03$$$$T^{19} +$$$$30\!\cdots\!21$$$$T^{20} +$$$$75\!\cdots\!66$$$$T^{21} -$$$$71\!\cdots\!81$$$$T^{22} -$$$$19\!\cdots\!01$$$$T^{23} +$$$$12\!\cdots\!81$$$$T^{24}$$
$89$ $$1 + 6366 T + 18939837 T^{2} + 33490901356 T^{3} + 36794607216366 T^{4} + 23062320055435131 T^{5} + 6331029765277091109 T^{6} +$$$$65\!\cdots\!65$$$$T^{7} +$$$$23\!\cdots\!84$$$$T^{8} +$$$$34\!\cdots\!44$$$$T^{9} +$$$$26\!\cdots\!92$$$$T^{10} +$$$$82\!\cdots\!30$$$$T^{11} +$$$$98\!\cdots\!30$$$$T^{12} +$$$$58\!\cdots\!70$$$$T^{13} +$$$$13\!\cdots\!12$$$$T^{14} +$$$$12\!\cdots\!96$$$$T^{15} +$$$$57\!\cdots\!64$$$$T^{16} +$$$$11\!\cdots\!85$$$$T^{17} +$$$$77\!\cdots\!29$$$$T^{18} +$$$$19\!\cdots\!59$$$$T^{19} +$$$$22\!\cdots\!06$$$$T^{20} +$$$$14\!\cdots\!24$$$$T^{21} +$$$$57\!\cdots\!37$$$$T^{22} +$$$$13\!\cdots\!54$$$$T^{23} +$$$$15\!\cdots\!61$$$$T^{24}$$
$97$ $$1 + 8052 T + 31462761 T^{2} + 80299670389 T^{3} + 153146011957221 T^{4} + 239405956107270819 T^{5} +$$$$33\!\cdots\!19$$$$T^{6} +$$$$42\!\cdots\!83$$$$T^{7} +$$$$52\!\cdots\!34$$$$T^{8} +$$$$60\!\cdots\!74$$$$T^{9} +$$$$65\!\cdots\!46$$$$T^{10} +$$$$66\!\cdots\!83$$$$T^{11} +$$$$64\!\cdots\!84$$$$T^{12} +$$$$60\!\cdots\!59$$$$T^{13} +$$$$54\!\cdots\!34$$$$T^{14} +$$$$45\!\cdots\!58$$$$T^{15} +$$$$36\!\cdots\!94$$$$T^{16} +$$$$27\!\cdots\!19$$$$T^{17} +$$$$19\!\cdots\!91$$$$T^{18} +$$$$12\!\cdots\!43$$$$T^{19} +$$$$73\!\cdots\!01$$$$T^{20} +$$$$35\!\cdots\!57$$$$T^{21} +$$$$12\!\cdots\!89$$$$T^{22} +$$$$29\!\cdots\!04$$$$T^{23} +$$$$33\!\cdots\!21$$$$T^{24}$$