# Properties

 Label 144.5.m.a Level $144$ Weight $5$ Character orbit 144.m Analytic conductor $14.885$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 144.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8852746841$$ Analytic rank: $$0$$ Dimension: $$14$$ Relative dimension: $$7$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + 11456 x^{6} - 19968 x^{5} + 31744 x^{4} - 139264 x^{3} + 491520 x^{2} - 1048576 x + 2097152$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{21}$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} + ( -1 + 2 \beta_{3} - \beta_{11} ) q^{4} + ( -\beta_{2} - \beta_{3} + \beta_{7} ) q^{5} + ( -1 - 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{7} + ( 6 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{8} +O(q^{10})$$ $$q -\beta_{4} q^{2} + ( -1 + 2 \beta_{3} - \beta_{11} ) q^{4} + ( -\beta_{2} - \beta_{3} + \beta_{7} ) q^{5} + ( -1 - 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{7} + ( 6 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{8} + ( -10 - 3 \beta_{1} - \beta_{2} - 10 \beta_{3} + 3 \beta_{4} + 5 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} ) q^{10} + ( -11 - 7 \beta_{1} + 2 \beta_{2} - 13 \beta_{3} + 11 \beta_{4} + \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{13} ) q^{11} + ( -11 - 18 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} ) q^{13} + ( -2 - \beta_{2} - 44 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 7 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 13 \beta_{10} - 3 \beta_{12} + \beta_{13} ) q^{14} + ( -6 + 10 \beta_{1} + 4 \beta_{2} - 54 \beta_{3} - 12 \beta_{4} - 6 \beta_{5} - 14 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} - 8 \beta_{10} - 4 \beta_{12} + 2 \beta_{13} ) q^{16} + ( 8 + 25 \beta_{1} - 3 \beta_{2} + 14 \beta_{3} - 29 \beta_{4} + 5 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} + 7 \beta_{8} - \beta_{9} - 4 \beta_{10} + 7 \beta_{11} - 2 \beta_{12} - 4 \beta_{13} ) q^{17} + ( -47 - 9 \beta_{1} - 6 \beta_{2} + 21 \beta_{3} + 47 \beta_{4} - 7 \beta_{5} + 3 \beta_{6} + 10 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} - 4 \beta_{13} ) q^{19} + ( -129 + 2 \beta_{1} - 2 \beta_{2} - 39 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 22 \beta_{10} - 5 \beta_{11} + 6 \beta_{12} - 2 \beta_{13} ) q^{20} + ( 65 + 8 \beta_{1} + 2 \beta_{2} - 147 \beta_{3} + 16 \beta_{4} + 8 \beta_{5} + 11 \beta_{6} + 4 \beta_{7} + 12 \beta_{8} + 5 \beta_{9} - 22 \beta_{10} + 4 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} ) q^{22} + ( -47 + 80 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} + 23 \beta_{4} - 13 \beta_{5} + 6 \beta_{6} + \beta_{7} + 6 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} - 7 \beta_{11} - 5 \beta_{12} - \beta_{13} ) q^{23} + ( 2 + 10 \beta_{1} + \beta_{3} - 98 \beta_{4} - 6 \beta_{5} + 12 \beta_{6} - 18 \beta_{8} - 4 \beta_{9} - 16 \beta_{10} - 8 \beta_{11} + 4 \beta_{12} - 8 \beta_{13} ) q^{25} + ( 238 + 13 \beta_{1} - 3 \beta_{2} + 146 \beta_{3} + 21 \beta_{4} + 10 \beta_{5} - 15 \beta_{6} + 18 \beta_{7} + 16 \beta_{8} + 11 \beta_{9} - 11 \beta_{10} - 14 \beta_{11} - \beta_{12} - 5 \beta_{13} ) q^{26} + ( -274 + 22 \beta_{1} - 10 \beta_{2} - 196 \beta_{3} + 14 \beta_{4} + 18 \beta_{5} + 2 \beta_{6} + 6 \beta_{7} - 10 \beta_{8} + 18 \beta_{9} - 6 \beta_{10} - 4 \beta_{11} + 6 \beta_{12} + 6 \beta_{13} ) q^{28} + ( -15 + 72 \beta_{1} + 4 \beta_{2} - 60 \beta_{3} + 28 \beta_{4} - 12 \beta_{5} + 4 \beta_{6} + 8 \beta_{7} - 24 \beta_{8} + 3 \beta_{9} - 28 \beta_{10} - 15 \beta_{11} + 12 \beta_{12} + 4 \beta_{13} ) q^{29} + ( 58 + 96 \beta_{1} + 20 \beta_{2} - 8 \beta_{3} + 38 \beta_{4} + 14 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} - 12 \beta_{8} - 8 \beta_{9} + 10 \beta_{10} + 30 \beta_{11} + 2 \beta_{12} + 10 \beta_{13} ) q^{31} + ( -224 + 50 \beta_{1} + 22 \beta_{2} + 500 \beta_{3} - 14 \beta_{4} + 18 \beta_{5} - 2 \beta_{6} - 10 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 18 \beta_{12} + 10 \beta_{13} ) q^{32} + ( 520 + 4 \beta_{1} - 6 \beta_{2} + 272 \beta_{3} - 18 \beta_{4} - 14 \beta_{5} + 24 \beta_{6} - 4 \beta_{7} - 10 \beta_{8} - 4 \beta_{9} - 10 \beta_{10} - 32 \beta_{11} - 14 \beta_{12} + 8 \beta_{13} ) q^{34} + ( -90 + 74 \beta_{1} - 32 \beta_{2} + 122 \beta_{3} - 74 \beta_{4} + 26 \beta_{5} - 2 \beta_{6} + 44 \beta_{7} - 2 \beta_{8} - 8 \beta_{9} - 8 \beta_{10} - 16 \beta_{11} - 8 \beta_{12} - 12 \beta_{13} ) q^{35} + ( -62 + 136 \beta_{1} - 7 \beta_{2} + 111 \beta_{3} + 58 \beta_{4} + 22 \beta_{5} - 70 \beta_{6} + 33 \beta_{7} + 16 \beta_{8} - 6 \beta_{9} - 6 \beta_{10} - 12 \beta_{11} - 6 \beta_{12} - 26 \beta_{13} ) q^{37} + ( -245 - 40 \beta_{1} - 13 \beta_{2} + 665 \beta_{3} + 32 \beta_{4} + 4 \beta_{5} + 6 \beta_{6} + 28 \beta_{7} - 24 \beta_{8} + 18 \beta_{9} - 29 \beta_{10} + 36 \beta_{11} + 5 \beta_{12} - 14 \beta_{13} ) q^{38} + ( -328 + 52 \beta_{1} - 14 \beta_{2} + 804 \beta_{3} + 110 \beta_{4} - 16 \beta_{5} - 68 \beta_{6} - 6 \beta_{7} + 6 \beta_{8} + 32 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} + 18 \beta_{12} + 16 \beta_{13} ) q^{40} + ( -10 - 76 \beta_{1} + 14 \beta_{2} - 34 \beta_{3} + 72 \beta_{4} + 76 \beta_{6} + 22 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} - 60 \beta_{10} + 10 \beta_{11} + 20 \beta_{12} - 4 \beta_{13} ) q^{41} + ( 140 - 4 \beta_{1} + 8 \beta_{2} + 148 \beta_{3} - 124 \beta_{4} - 20 \beta_{5} + 8 \beta_{6} + 16 \beta_{7} + 4 \beta_{8} - 40 \beta_{9} - \beta_{10} + 24 \beta_{11} + 16 \beta_{12} + 8 \beta_{13} ) q^{43} + ( 1103 + 180 \beta_{1} - 8 \beta_{2} + 303 \beta_{3} - 102 \beta_{4} - 20 \beta_{5} + 18 \beta_{6} + 5 \beta_{7} + 10 \beta_{8} + 10 \beta_{9} - 80 \beta_{10} + 5 \beta_{11} - 4 \beta_{12} + 34 \beta_{13} ) q^{44} + ( -358 - 8 \beta_{1} + 17 \beta_{2} + 1316 \beta_{3} - 36 \beta_{4} + 6 \beta_{5} + 5 \beta_{6} - 92 \beta_{7} + 6 \beta_{8} - 17 \beta_{9} - 107 \beta_{10} + 32 \beta_{11} - 5 \beta_{12} + 3 \beta_{13} ) q^{46} + ( 34 + 12 \beta_{1} + 28 \beta_{2} + 468 \beta_{3} - 50 \beta_{4} - 2 \beta_{5} - 26 \beta_{6} - 58 \beta_{7} - 48 \beta_{9} + 6 \beta_{10} + 18 \beta_{11} - 10 \beta_{12} - 10 \beta_{13} ) q^{47} + ( 79 + 102 \beta_{1} - 26 \beta_{2} - 4 \beta_{3} - 22 \beta_{4} - 10 \beta_{5} + 100 \beta_{6} + 2 \beta_{7} + 10 \beta_{8} - 14 \beta_{9} + 112 \beta_{10} + 26 \beta_{11} + 12 \beta_{12} ) q^{49} + ( -1448 + \beta_{1} - 28 \beta_{2} - 336 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} - 124 \beta_{6} - 32 \beta_{7} - 8 \beta_{8} + 12 \beta_{9} - 44 \beta_{10} - 96 \beta_{11} + 12 \beta_{12} - 4 \beta_{13} ) q^{50} + ( 1345 - 142 \beta_{1} + 30 \beta_{2} + 641 \beta_{3} - 282 \beta_{4} - 18 \beta_{5} + 30 \beta_{6} - \beta_{7} - 2 \beta_{8} + 38 \beta_{9} - 118 \beta_{10} - 35 \beta_{11} - 18 \beta_{12} + 2 \beta_{13} ) q^{52} + ( 134 + 156 \beta_{1} - 13 \beta_{2} - 75 \beta_{3} + 42 \beta_{4} + 22 \beta_{5} - 202 \beta_{6} + 19 \beta_{7} + 12 \beta_{8} + 14 \beta_{9} + 14 \beta_{10} + 28 \beta_{11} + 14 \beta_{12} - 6 \beta_{13} ) q^{53} + ( -701 + 210 \beta_{1} - 40 \beta_{2} - 68 \beta_{3} + 101 \beta_{4} - 35 \beta_{5} + 32 \beta_{6} + 65 \beta_{7} + 12 \beta_{8} - 48 \beta_{9} + 24 \beta_{10} + 49 \beta_{11} - 17 \beta_{12} - 9 \beta_{13} ) q^{55} + ( 552 + 126 \beta_{1} + 26 \beta_{2} - 1784 \beta_{3} + 322 \beta_{4} - 42 \beta_{5} + 118 \beta_{6} - 38 \beta_{7} + 14 \beta_{8} + 54 \beta_{9} + 30 \beta_{10} - 42 \beta_{11} + 26 \beta_{12} + 26 \beta_{13} ) q^{56} + ( -1454 + 43 \beta_{1} - 11 \beta_{2} - 658 \beta_{3} + 59 \beta_{4} + 18 \beta_{5} - 131 \beta_{6} - 110 \beta_{7} + 48 \beta_{8} + 63 \beta_{9} - 3 \beta_{10} + 18 \beta_{11} + 7 \beta_{12} - \beta_{13} ) q^{58} + ( 242 + 114 \beta_{1} + 32 \beta_{2} + 330 \beta_{3} - 406 \beta_{4} - 66 \beta_{5} + 32 \beta_{6} + 64 \beta_{7} + 30 \beta_{8} - 56 \beta_{9} + 13 \beta_{10} + 68 \beta_{11} - 12 \beta_{12} + 32 \beta_{13} ) q^{59} + ( -199 + 202 \beta_{1} + 34 \beta_{2} - 362 \beta_{3} + 180 \beta_{4} - 28 \beta_{5} + 34 \beta_{6} + 68 \beta_{7} - 74 \beta_{8} + 21 \beta_{9} + 294 \beta_{10} - 31 \beta_{11} + 10 \beta_{12} + 34 \beta_{13} ) q^{61} + ( 856 + 104 \beta_{1} - 36 \beta_{2} - 1624 \beta_{3} + 24 \beta_{4} + 36 \beta_{5} - 8 \beta_{6} - 96 \beta_{7} + 92 \beta_{8} - 64 \beta_{9} + 36 \beta_{10} + 88 \beta_{11} - 52 \beta_{12} + 8 \beta_{13} ) q^{62} + ( 924 - 512 \beta_{1} + 52 \beta_{2} - 1636 \beta_{3} + 212 \beta_{4} + 64 \beta_{5} + 64 \beta_{6} - 80 \beta_{7} + 12 \beta_{8} - 56 \beta_{9} - 92 \beta_{10} + 4 \beta_{11} - 28 \beta_{12} + 24 \beta_{13} ) q^{64} + ( 204 - 104 \beta_{1} - 62 \beta_{2} - 214 \beta_{3} + 484 \beta_{4} - 12 \beta_{5} + 180 \beta_{6} + 26 \beta_{7} - 64 \beta_{8} - 14 \beta_{9} + 228 \beta_{10} + 122 \beta_{11} - 12 \beta_{12} - 60 \beta_{13} ) q^{65} + ( 643 + 341 \beta_{1} + 10 \beta_{2} - 365 \beta_{3} - 223 \beta_{4} + 119 \beta_{5} + 7 \beta_{6} + 78 \beta_{7} - \beta_{8} - 42 \beta_{9} - 42 \beta_{10} - 84 \beta_{11} - 42 \beta_{12} - 88 \beta_{13} ) q^{67} + ( -1554 - 448 \beta_{1} + 40 \beta_{2} - 624 \beta_{3} - 516 \beta_{4} + 72 \beta_{5} + 20 \beta_{6} - 40 \beta_{7} - 92 \beta_{8} + 52 \beta_{9} + 8 \beta_{10} - 34 \beta_{11} + 80 \beta_{12} - 12 \beta_{13} ) q^{68} + ( 946 - 232 \beta_{1} - 38 \beta_{2} - 1674 \beta_{3} + 200 \beta_{4} + 24 \beta_{5} + 76 \beta_{6} - 8 \beta_{7} - 112 \beta_{8} + 100 \beta_{9} - 38 \beta_{10} - 184 \beta_{11} + 22 \beta_{12} - 60 \beta_{13} ) q^{70} + ( -1807 - 608 \beta_{1} + 72 \beta_{2} - 14 \beta_{3} - 73 \beta_{4} + 51 \beta_{5} + 150 \beta_{6} + 81 \beta_{7} - 42 \beta_{8} - 28 \beta_{9} + 50 \beta_{10} - 119 \beta_{11} - 53 \beta_{12} + 47 \beta_{13} ) q^{71} + ( -52 - 409 \beta_{1} + 83 \beta_{2} + 68 \beta_{3} + 367 \beta_{4} - 7 \beta_{5} - 208 \beta_{6} + 27 \beta_{7} + 57 \beta_{8} - 37 \beta_{9} + 254 \beta_{10} + 65 \beta_{11} + 64 \beta_{12} - 18 \beta_{13} ) q^{73} + ( 1714 - 37 \beta_{1} + 23 \beta_{2} + 498 \beta_{3} + 13 \beta_{4} - 16 \beta_{5} - 79 \beta_{6} - 86 \beta_{7} - 50 \beta_{8} + 7 \beta_{9} - 181 \beta_{10} - 10 \beta_{11} - 103 \beta_{12} - 45 \beta_{13} ) q^{74} + ( -2015 - 694 \beta_{1} - 58 \beta_{2} - 1153 \beta_{3} + 204 \beta_{4} - 122 \beta_{5} + 176 \beta_{6} + 75 \beta_{7} + 44 \beta_{8} + 24 \beta_{9} - 126 \beta_{10} + 5 \beta_{11} - 46 \beta_{12} + 68 \beta_{13} ) q^{76} + ( 384 - 530 \beta_{1} - 66 \beta_{2} + 570 \beta_{3} - 28 \beta_{4} + 68 \beta_{5} - 66 \beta_{6} - 132 \beta_{7} + 50 \beta_{8} + 2 \beta_{9} + 330 \beta_{10} - 104 \beta_{11} + 102 \beta_{12} - 66 \beta_{13} ) q^{77} + ( -184 - 348 \beta_{1} + 8 \beta_{2} - 1948 \beta_{3} - 648 \beta_{4} - 112 \beta_{5} - 148 \beta_{6} - 76 \beta_{7} + 4 \beta_{8} - 40 \beta_{9} + 116 \beta_{10} + 12 \beta_{11} - 36 \beta_{12} + 4 \beta_{13} ) q^{79} + ( -476 - 882 \beta_{1} + 114 \beta_{2} + 1232 \beta_{3} + 318 \beta_{4} - 90 \beta_{5} + 194 \beta_{6} - 66 \beta_{7} + 58 \beta_{8} + 66 \beta_{9} + 230 \beta_{10} + 90 \beta_{11} - 22 \beta_{12} + 54 \beta_{13} ) q^{80} + ( 1240 + 216 \beta_{1} - 148 \beta_{2} + 640 \beta_{3} + 32 \beta_{4} - 12 \beta_{5} - 208 \beta_{6} + 96 \beta_{7} + 132 \beta_{8} + 56 \beta_{9} - 172 \beta_{10} + 72 \beta_{11} + 12 \beta_{12} + 32 \beta_{13} ) q^{82} + ( 1300 + 108 \beta_{1} - 20 \beta_{2} - 1472 \beta_{3} + 424 \beta_{4} - 104 \beta_{5} + 21 \beta_{6} - 40 \beta_{7} + 100 \beta_{8} - 36 \beta_{9} - 36 \beta_{10} - 72 \beta_{11} - 36 \beta_{12} + 60 \beta_{13} ) q^{83} + ( 434 - 544 \beta_{1} - 14 \beta_{2} - 520 \beta_{3} - 382 \beta_{4} + 46 \beta_{5} + 394 \beta_{6} + 88 \beta_{7} - 136 \beta_{8} + 26 \beta_{9} + 26 \beta_{10} + 52 \beta_{11} + 26 \beta_{12} - 74 \beta_{13} ) q^{85} + ( 271 + 32 \beta_{1} - 152 \beta_{2} + 1579 \beta_{3} + 64 \beta_{4} + 128 \beta_{5} - 265 \beta_{6} + 80 \beta_{7} + 80 \beta_{8} + 9 \beta_{9} + 200 \beta_{10} - 48 \beta_{11} - 40 \beta_{12} - 87 \beta_{13} ) q^{86} + ( -44 - 472 \beta_{1} + 34 \beta_{2} + 556 \beta_{3} - 1014 \beta_{4} + 68 \beta_{5} + 196 \beta_{6} - 162 \beta_{7} + 18 \beta_{8} + 160 \beta_{9} + 202 \beta_{10} - 100 \beta_{11} + 58 \beta_{12} + 104 \beta_{13} ) q^{88} + ( -92 - 193 \beta_{1} - 13 \beta_{2} - 420 \beta_{3} - 457 \beta_{4} - 47 \beta_{5} - 256 \beta_{6} + 187 \beta_{7} - 127 \beta_{8} + 75 \beta_{9} + 318 \beta_{10} - 63 \beta_{11} + 112 \beta_{12} - 50 \beta_{13} ) q^{89} + ( -2124 - 468 \beta_{1} + 28 \beta_{2} - 2176 \beta_{3} + 144 \beta_{4} + 48 \beta_{5} + 28 \beta_{6} + 56 \beta_{7} + 20 \beta_{8} - 108 \beta_{9} - 30 \beta_{10} + 64 \beta_{11} + 44 \beta_{12} + 28 \beta_{13} ) q^{91} + ( 602 - 1198 \beta_{1} + 34 \beta_{2} - 1108 \beta_{3} + 306 \beta_{4} - 42 \beta_{5} - 226 \beta_{6} - 46 \beta_{7} - 38 \beta_{8} - 50 \beta_{9} + 270 \beta_{10} + 116 \beta_{11} + 34 \beta_{12} + 202 \beta_{13} ) q^{92} + ( 24 - 264 \beta_{1} - 20 \beta_{2} - 8 \beta_{3} - 40 \beta_{4} + 76 \beta_{5} - 368 \beta_{6} - 32 \beta_{7} - 44 \beta_{8} - 168 \beta_{9} + 132 \beta_{10} + 104 \beta_{11} - 52 \beta_{12} - 16 \beta_{13} ) q^{94} + ( 216 + 345 \beta_{1} + 64 \beta_{2} - 2697 \beta_{3} - 278 \beta_{4} + 24 \beta_{5} - 96 \beta_{6} - 289 \beta_{7} - 23 \beta_{8} - 208 \beta_{9} - 48 \beta_{10} + \beta_{11} - 81 \beta_{12} - 63 \beta_{13} ) q^{95} + ( 608 + 819 \beta_{1} - 201 \beta_{2} - 78 \beta_{3} + 25 \beta_{4} - 81 \beta_{5} - 330 \beta_{6} + \beta_{7} + 61 \beta_{8} - 83 \beta_{9} - 212 \beta_{10} + 237 \beta_{11} + 82 \beta_{12} - 36 \beta_{13} ) q^{97} + ( 976 + 24 \beta_{1} - 180 \beta_{2} + 1600 \beta_{3} + 37 \beta_{4} - 20 \beta_{5} + 128 \beta_{6} - 24 \beta_{7} - 28 \beta_{8} - 72 \beta_{9} + 180 \beta_{10} + 124 \beta_{12} - 128 \beta_{13} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q + 2q^{2} - 8q^{4} + 2q^{5} - 4q^{7} + 92q^{8} + O(q^{10})$$ $$14q + 2q^{2} - 8q^{4} + 2q^{5} - 4q^{7} + 92q^{8} - 100q^{10} - 94q^{11} - 2q^{13} - 44q^{14} - 168q^{16} + 4q^{17} - 706q^{19} - 1900q^{20} + 900q^{22} - 1148q^{23} + 3416q^{26} - 3784q^{28} - 862q^{29} - 3208q^{32} + 7508q^{34} - 1340q^{35} - 1826q^{37} - 3568q^{38} - 5144q^{40} + 1694q^{43} + 14636q^{44} - 5316q^{46} + 682q^{49} - 20070q^{50} + 20452q^{52} + 482q^{53} - 11780q^{55} + 6952q^{56} - 20456q^{58} + 2786q^{59} - 3778q^{61} + 11472q^{62} + 15808q^{64} + 2020q^{65} + 7998q^{67} - 18032q^{68} + 15296q^{70} - 19964q^{71} + 23780q^{74} - 23996q^{76} + 9508q^{77} - 1384q^{80} + 16016q^{82} + 17282q^{83} + 9948q^{85} + 4796q^{86} + 7288q^{88} - 28036q^{91} + 14632q^{92} + 432q^{94} - 4q^{97} + 12246q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + 11456 x^{6} - 19968 x^{5} + 31744 x^{4} - 139264 x^{3} + 491520 x^{2} - 1048576 x + 2097152$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2291 \nu^{13} - 13836 \nu^{12} + 73187 \nu^{11} - 134754 \nu^{10} - 13546 \nu^{9} + 515160 \nu^{8} + 2711160 \nu^{7} - 25589280 \nu^{6} + 58745024 \nu^{5} - 46252032 \nu^{4} - 25039872 \nu^{3} - 96387072 \nu^{2} + 2307620864 \nu - 8127774720$$$$)/ 678952960$$ $$\beta_{2}$$ $$=$$ $$($$$$2559 \nu^{13} + 209196 \nu^{12} - 627631 \nu^{11} + 1042042 \nu^{10} - 73358 \nu^{9} + 4773352 \nu^{8} - 52296280 \nu^{7} + 214229280 \nu^{6} - 430709696 \nu^{5} + 72770560 \nu^{4} - 10267648 \nu^{3} + 3766566912 \nu^{2} - 21547548672 \nu + 48311304192$$$$)/ 678952960$$ $$\beta_{3}$$ $$=$$ $$($$$$2875 \nu^{13} - 13444 \nu^{12} + 26581 \nu^{11} - 16062 \nu^{10} + 24954 \nu^{9} - 748984 \nu^{8} + 4619080 \nu^{7} - 10623840 \nu^{6} + 11499840 \nu^{5} - 5960704 \nu^{4} + 51930112 \nu^{3} - 429211648 \nu^{2} + 1316257792 \nu - 1279524864$$$$)/ 678952960$$ $$\beta_{4}$$ $$=$$ $$($$$$7471 \nu^{13} - 6884 \nu^{12} + 4513 \nu^{11} - 41366 \nu^{10} + 334706 \nu^{9} - 2131320 \nu^{8} + 4706600 \nu^{7} - 103520 \nu^{6} + 597056 \nu^{5} - 57182208 \nu^{4} + 189473792 \nu^{3} - 625000448 \nu^{2} + 238452736 \nu + 2696151040$$$$)/ 678952960$$ $$\beta_{5}$$ $$=$$ $$($$$$-739 \nu^{13} + 1452 \nu^{12} - 2221 \nu^{11} + 4102 \nu^{10} - 25690 \nu^{9} + 205736 \nu^{8} - 666120 \nu^{7} + 766240 \nu^{6} - 864064 \nu^{5} + 4510208 \nu^{4} - 17243136 \nu^{3} + 75300864 \nu^{2} + 276922368 \nu - 198180864$$$$)/48496640$$ $$\beta_{6}$$ $$=$$ $$($$$$-11609 \nu^{13} + 81068 \nu^{12} - 216535 \nu^{11} + 92170 \nu^{10} - 445086 \nu^{9} + 4629992 \nu^{8} - 22287000 \nu^{7} + 63665440 \nu^{6} - 81223104 \nu^{5} + 58993664 \nu^{4} - 414198784 \nu^{3} + 2235252736 \nu^{2} - 5795840000 \nu + 7904428032$$$$)/ 678952960$$ $$\beta_{7}$$ $$=$$ $$($$$$-17845 \nu^{13} + 27996 \nu^{12} - 118459 \nu^{11} + 66818 \nu^{10} + 149274 \nu^{9} + 3146376 \nu^{8} - 12122680 \nu^{7} + 32805280 \nu^{6} - 20326080 \nu^{5} - 66960384 \nu^{4} + 221668352 \nu^{3} + 1140867072 \nu^{2} - 3096936448 \nu + 1436286976$$$$)/ 678952960$$ $$\beta_{8}$$ $$=$$ $$($$$$-19011 \nu^{13} - 91628 \nu^{12} + 527155 \nu^{11} - 1059490 \nu^{10} + 1147766 \nu^{9} + 4999448 \nu^{8} + 17139960 \nu^{7} - 124652320 \nu^{6} + 329833664 \nu^{5} - 256497664 \nu^{4} - 309441536 \nu^{3} + 179912704 \nu^{2} + 8940912640 \nu - 36868718592$$$$)/ 678952960$$ $$\beta_{9}$$ $$=$$ $$($$$$28401 \nu^{13} - 32588 \nu^{12} - 184641 \nu^{11} + 518662 \nu^{10} - 277010 \nu^{9} - 2018984 \nu^{8} - 431400 \nu^{7} + 42427360 \nu^{6} - 178329664 \nu^{5} + 148231168 \nu^{4} - 371536896 \nu^{3} - 414662656 \nu^{2} - 5578063872 \nu + 19838795776$$$$)/ 678952960$$ $$\beta_{10}$$ $$=$$ $$($$$$30299 \nu^{13} - 175428 \nu^{12} + 396085 \nu^{11} - 537790 \nu^{10} + 1159226 \nu^{9} - 9651512 \nu^{8} + 53625160 \nu^{7} - 150852960 \nu^{6} + 203316544 \nu^{5} - 134759424 \nu^{4} + 982549504 \nu^{3} - 4972855296 \nu^{2} + 16046653440 \nu - 20694433792$$$$)/ 678952960$$ $$\beta_{11}$$ $$=$$ $$($$$$-8885 \nu^{13} + 54092 \nu^{12} - 127643 \nu^{11} + 205586 \nu^{10} - 440742 \nu^{9} + 3542632 \nu^{8} - 17765240 \nu^{7} + 49878560 \nu^{6} - 61533120 \nu^{5} + 86843392 \nu^{4} - 323572736 \nu^{3} + 1651974144 \nu^{2} - 5472813056 \nu + 5876088832$$$$)/ 169738240$$ $$\beta_{12}$$ $$=$$ $$($$$$-73363 \nu^{13} + 121444 \nu^{12} + 210563 \nu^{11} - 703506 \nu^{10} - 613450 \nu^{9} + 15576312 \nu^{8} - 36025480 \nu^{7} - 13428640 \nu^{6} + 250450112 \nu^{5} - 165837824 \nu^{4} - 888906752 \nu^{3} + 2383659008 \nu^{2} - 142639104 \nu - 37401919488$$$$)/ 678952960$$ $$\beta_{13}$$ $$=$$ $$($$$$-74425 \nu^{13} + 92076 \nu^{12} + 124041 \nu^{11} - 284342 \nu^{10} + 496674 \nu^{9} + 12413416 \nu^{8} - 28455320 \nu^{7} - 40588000 \nu^{6} + 293167680 \nu^{5} - 121549824 \nu^{4} - 725675008 \nu^{3} + 2751741952 \nu^{2} + 4283072512 \nu - 48499523584$$$$)/ 678952960$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{3} + 2$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{12} - 2 \beta_{10} - 2 \beta_{9} + \beta_{8} - 2 \beta_{4} + 4 \beta_{3} + 3 \beta_{1} - 7$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{13} + 4 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} - \beta_{8} + 4 \beta_{7} - 2 \beta_{6} - 6 \beta_{4} - 16 \beta_{3} + 2 \beta_{2} + 9 \beta_{1} - 1$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{13} + 12 \beta_{11} + 32 \beta_{10} - 4 \beta_{9} - \beta_{8} - 12 \beta_{7} + 10 \beta_{6} - 26 \beta_{4} - 116 \beta_{3} - 2 \beta_{2} - 59 \beta_{1} - 5$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$10 \beta_{13} - 8 \beta_{12} + 52 \beta_{11} + 24 \beta_{10} - 4 \beta_{9} - 7 \beta_{8} + 12 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} + 160 \beta_{4} + 162 \beta_{3} - 18 \beta_{2} + 79 \beta_{1} + 765$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-26 \beta_{13} + 12 \beta_{12} + 84 \beta_{11} - 52 \beta_{10} - 16 \beta_{9} + 35 \beta_{8} + 76 \beta_{7} - 66 \beta_{6} + 104 \beta_{5} + 454 \beta_{4} + 860 \beta_{3} - 54 \beta_{2} - 247 \beta_{1} - 1585$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$34 \beta_{13} - 136 \beta_{12} - 28 \beta_{11} - 344 \beta_{10} - 260 \beta_{9} + 185 \beta_{8} + 44 \beta_{7} + 330 \beta_{6} - 270 \beta_{5} - 416 \beta_{4} + 5506 \beta_{3} + 22 \beta_{2} - 841 \beta_{1} - 1579$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$-74 \beta_{13} + 268 \beta_{12} + 516 \beta_{11} + 1100 \beta_{10} + 224 \beta_{9} + 343 \beta_{8} + 412 \beta_{7} + 814 \beta_{6} - 60 \beta_{5} - 1102 \beta_{4} + 8000 \beta_{3} + 586 \beta_{2} + 333 \beta_{1} + 8371$$$$)/8$$ $$\nu^{9}$$ $$=$$ $$($$$$1946 \beta_{13} - 528 \beta_{12} - 380 \beta_{11} + 2848 \beta_{10} + 1364 \beta_{9} + 1437 \beta_{8} - 1524 \beta_{7} + 1154 \beta_{6} + 614 \beta_{5} + 532 \beta_{4} - 15042 \beta_{3} + 1134 \beta_{2} - 10389 \beta_{1} + 4841$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$4046 \beta_{13} - 3372 \beta_{12} + 3460 \beta_{11} - 5708 \beta_{10} - 248 \beta_{9} - 157 \beta_{8} + 700 \beta_{7} - 13850 \beta_{6} - 320 \beta_{5} + 2910 \beta_{4} + 73348 \beta_{3} + 2914 \beta_{2} + 16569 \beta_{1} + 84351$$$$)/8$$ $$\nu^{11}$$ $$=$$ $$($$$$-7230 \beta_{13} + 3624 \beta_{12} + 4260 \beta_{11} - 13896 \beta_{10} - 10932 \beta_{9} + 9033 \beta_{8} - 12724 \beta_{7} - 49206 \beta_{6} + 13506 \beta_{5} + 1984 \beta_{4} + 10978 \beta_{3} + 5686 \beta_{2} - 27465 \beta_{1} - 143499$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$($$$$15734 \beta_{13} - 5300 \beta_{12} - 8476 \beta_{11} + 1484 \beta_{10} - 56128 \beta_{9} - 1297 \beta_{8} - 88068 \beta_{7} + 52590 \beta_{6} - 40244 \beta_{5} - 79110 \beta_{4} + 378904 \beta_{3} + 39018 \beta_{2} - 13019 \beta_{1} + 245835$$$$)/8$$ $$\nu^{13}$$ $$=$$ $$($$$$-23574 \beta_{13} - 6464 \beta_{12} + 178052 \beta_{11} + 253168 \beta_{10} + 23716 \beta_{9} + 11813 \beta_{8} - 104692 \beta_{7} + 100306 \beta_{6} + 48462 \beta_{5} + 340348 \beta_{4} + 55670 \beta_{3} + 85822 \beta_{2} + 747987 \beta_{1} + 313681$$$$)/8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −2.40693 + 1.48549i 0.336831 + 2.80830i 1.03712 + 2.63142i −2.15805 − 1.82834i 2.79265 + 0.448449i 0.153862 − 2.82424i 2.24452 − 1.72109i −2.40693 − 1.48549i 0.336831 − 2.80830i 1.03712 − 2.63142i −2.15805 + 1.82834i 2.79265 − 0.448449i 0.153862 + 2.82424i 2.24452 + 1.72109i
−3.89242 + 0.921438i 0 14.3019 7.17325i 8.04297 8.04297i 0 −49.8797 −49.0594 + 41.0996i 0 −23.8955 + 38.7177i
19.2 −2.47147 3.14513i 0 −3.78368 + 15.5462i −27.2309 + 27.2309i 0 50.3097 58.2460 26.5217i 0 152.945 + 18.3444i
19.3 −1.59430 3.66854i 0 −10.9164 + 11.6975i 14.6016 14.6016i 0 −24.0210 60.3169 + 21.3980i 0 −76.8459 30.2872i
19.4 −0.329715 + 3.98639i 0 −15.7826 2.62875i −4.72348 + 4.72348i 0 45.3712 15.6830 62.0487i 0 −17.2722 20.3870i
19.5 2.34420 3.24110i 0 −5.00945 15.1956i 29.2002 29.2002i 0 59.6196 −60.9935 19.3854i 0 −26.1896 163.092i
19.6 2.97810 + 2.67038i 0 1.73818 + 15.9053i 2.84710 2.84710i 0 −76.7794 −37.2967 + 52.0092i 0 16.0818 0.876123i
19.7 3.96560 0.523430i 0 15.4520 4.15143i −21.7374 + 21.7374i 0 −6.62054 59.1037 24.5510i 0 −74.8239 + 97.5799i
91.1 −3.89242 0.921438i 0 14.3019 + 7.17325i 8.04297 + 8.04297i 0 −49.8797 −49.0594 41.0996i 0 −23.8955 38.7177i
91.2 −2.47147 + 3.14513i 0 −3.78368 15.5462i −27.2309 27.2309i 0 50.3097 58.2460 + 26.5217i 0 152.945 18.3444i
91.3 −1.59430 + 3.66854i 0 −10.9164 11.6975i 14.6016 + 14.6016i 0 −24.0210 60.3169 21.3980i 0 −76.8459 + 30.2872i
91.4 −0.329715 3.98639i 0 −15.7826 + 2.62875i −4.72348 4.72348i 0 45.3712 15.6830 + 62.0487i 0 −17.2722 + 20.3870i
91.5 2.34420 + 3.24110i 0 −5.00945 + 15.1956i 29.2002 + 29.2002i 0 59.6196 −60.9935 + 19.3854i 0 −26.1896 + 163.092i
91.6 2.97810 2.67038i 0 1.73818 15.9053i 2.84710 + 2.84710i 0 −76.7794 −37.2967 52.0092i 0 16.0818 + 0.876123i
91.7 3.96560 + 0.523430i 0 15.4520 + 4.15143i −21.7374 21.7374i 0 −6.62054 59.1037 + 24.5510i 0 −74.8239 97.5799i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.7 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.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.5.m.a 14
3.b odd 2 1 16.5.f.a 14
4.b odd 2 1 576.5.m.a 14
12.b even 2 1 64.5.f.a 14
16.e even 4 1 576.5.m.a 14
16.f odd 4 1 inner 144.5.m.a 14
24.f even 2 1 128.5.f.a 14
24.h odd 2 1 128.5.f.b 14
48.i odd 4 1 64.5.f.a 14
48.i odd 4 1 128.5.f.a 14
48.k even 4 1 16.5.f.a 14
48.k even 4 1 128.5.f.b 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.5.f.a 14 3.b odd 2 1
16.5.f.a 14 48.k even 4 1
64.5.f.a 14 12.b even 2 1
64.5.f.a 14 48.i odd 4 1
128.5.f.a 14 24.f even 2 1
128.5.f.a 14 48.i odd 4 1
128.5.f.b 14 24.h odd 2 1
128.5.f.b 14 48.k even 4 1
144.5.m.a 14 1.a even 1 1 trivial
144.5.m.a 14 16.f odd 4 1 inner
576.5.m.a 14 4.b odd 2 1
576.5.m.a 14 16.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$13\!\cdots\!64$$$$T_{5}^{6} -$$$$14\!\cdots\!16$$$$T_{5}^{5} +$$$$72\!\cdots\!04$$$$T_{5}^{4} +$$$$20\!\cdots\!68$$$$T_{5}^{3} +$$$$38\!\cdots\!00$$$$T_{5}^{2} -$$$$27\!\cdots\!00$$$$T_{5} +$$$$95\!\cdots\!08$$">$$T_{5}^{14} - \cdots$$ acting on $$S_{5}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$268435456 - 33554432 T + 6291456 T^{2} - 2621440 T^{3} + 491520 T^{4} + 90112 T^{5} - 46080 T^{6} - 2560 T^{7} - 2880 T^{8} + 352 T^{9} + 120 T^{10} - 40 T^{11} + 6 T^{12} - 2 T^{13} + T^{14}$$
$3$ $$T^{14}$$
$5$ $$95385721607700608 - 27076575211548800 T + 3843032860840000 T^{2} + 208330635859968 T^{3} + 72061290078304 T^{4} - 14025915627616 T^{5} + 1305653282864 T^{6} - 19416658944 T^{7} + 3101720 T^{8} - 2850584 T^{9} + 3106060 T^{10} - 2688 T^{11} + 2 T^{12} - 2 T^{13} + T^{14}$$
$7$ $$( -82884464768 - 13846485056 T - 67418144 T^{2} + 20706288 T^{3} + 35592 T^{4} - 8572 T^{5} + 2 T^{6} + T^{7} )^{2}$$
$11$ $$47\!\cdots\!12$$$$-$$$$30\!\cdots\!92$$$$T +$$$$95\!\cdots\!36$$$$T^{2} -$$$$65\!\cdots\!84$$$$T^{3} +$$$$20\!\cdots\!48$$$$T^{4} - 2830209232291459680 T^{5} + 219305901738719280 T^{6} - 1162127446795776 T^{7} + 2291939461144 T^{8} + 32065636328 T^{9} + 784133132 T^{10} - 2341824 T^{11} + 4418 T^{12} + 94 T^{13} + T^{14}$$
$13$ $$24\!\cdots\!52$$$$-$$$$17\!\cdots\!12$$$$T +$$$$62\!\cdots\!36$$$$T^{2} -$$$$78\!\cdots\!80$$$$T^{3} +$$$$49\!\cdots\!08$$$$T^{4} - 13395006265199179680 T^{5} + 938835469167525936 T^{6} - 8456611120356352 T^{7} + 24609579825688 T^{8} + 662386022680 T^{9} + 6547368972 T^{10} + 6826112 T^{11} + 2 T^{12} + 2 T^{13} + T^{14}$$
$17$ $$( 12009518203797632 - 119000879480896 T - 1432677679200 T^{2} + 14701879344 T^{3} + 13778968 T^{4} - 250892 T^{5} - 2 T^{6} + T^{7} )^{2}$$
$19$ $$37\!\cdots\!48$$$$-$$$$19\!\cdots\!84$$$$T +$$$$52\!\cdots\!36$$$$T^{2} +$$$$10\!\cdots\!20$$$$T^{3} +$$$$87\!\cdots\!60$$$$T^{4} +$$$$29\!\cdots\!48$$$$T^{5} +$$$$54\!\cdots\!96$$$$T^{6} + 1364277329112460800 T^{7} + 15308192398571800 T^{8} + 46983788219800 T^{9} + 71849973004 T^{10} + 9374912 T^{11} + 249218 T^{12} + 706 T^{13} + T^{14}$$
$23$ $$( -54911897542109056 - 1959672332215360 T + 7489328380960 T^{2} + 215281147888 T^{3} - 327850376 T^{4} - 934844 T^{5} + 574 T^{6} + T^{7} )^{2}$$
$29$ $$27\!\cdots\!28$$$$+$$$$58\!\cdots\!72$$$$T +$$$$63\!\cdots\!64$$$$T^{2} +$$$$36\!\cdots\!92$$$$T^{3} +$$$$12\!\cdots\!92$$$$T^{4} +$$$$25\!\cdots\!48$$$$T^{5} +$$$$35\!\cdots\!00$$$$T^{6} +$$$$63\!\cdots\!52$$$$T^{7} + 2138874201321723160 T^{8} + 4180270176993128 T^{9} + 4196992328204 T^{10} + 435330432 T^{11} + 371522 T^{12} + 862 T^{13} + T^{14}$$
$31$ $$71\!\cdots\!96$$$$+$$$$40\!\cdots\!52$$$$T^{2} +$$$$76\!\cdots\!80$$$$T^{4} +$$$$57\!\cdots\!84$$$$T^{6} + 13297300404014415872 T^{8} + 13339923865600 T^{10} + 6024960 T^{12} + T^{14}$$
$37$ $$51\!\cdots\!68$$$$-$$$$19\!\cdots\!88$$$$T +$$$$35\!\cdots\!04$$$$T^{2} +$$$$35\!\cdots\!92$$$$T^{3} +$$$$15\!\cdots\!28$$$$T^{4} +$$$$12\!\cdots\!88$$$$T^{5} +$$$$64\!\cdots\!32$$$$T^{6} +$$$$10\!\cdots\!64$$$$T^{7} + 37374659035443657496 T^{8} + 35578094339272856 T^{9} + 17274822237964 T^{10} + 1554716288 T^{11} + 1667138 T^{12} + 1826 T^{13} + T^{14}$$
$41$ $$24\!\cdots\!04$$$$+$$$$51\!\cdots\!76$$$$T^{2} +$$$$20\!\cdots\!12$$$$T^{4} +$$$$31\!\cdots\!48$$$$T^{6} +$$$$23\!\cdots\!56$$$$T^{8} + 85694612004864 T^{10} + 15036672 T^{12} + T^{14}$$
$43$ $$11\!\cdots\!68$$$$-$$$$68\!\cdots\!60$$$$T +$$$$20\!\cdots\!00$$$$T^{2} -$$$$88\!\cdots\!68$$$$T^{3} +$$$$18\!\cdots\!68$$$$T^{4} -$$$$15\!\cdots\!16$$$$T^{5} +$$$$42\!\cdots\!04$$$$T^{6} -$$$$13\!\cdots\!48$$$$T^{7} +$$$$19\!\cdots\!00$$$$T^{8} - 142596531803367912 T^{9} + 57605028643980 T^{10} - 8486946368 T^{11} + 1434818 T^{12} - 1694 T^{13} + T^{14}$$
$47$ $$12\!\cdots\!76$$$$+$$$$10\!\cdots\!80$$$$T^{2} +$$$$38\!\cdots\!80$$$$T^{4} +$$$$52\!\cdots\!84$$$$T^{6} +$$$$32\!\cdots\!52$$$$T^{8} + 104964818558976 T^{10} + 16427776 T^{12} + T^{14}$$
$53$ $$29\!\cdots\!88$$$$+$$$$53\!\cdots\!48$$$$T +$$$$48\!\cdots\!04$$$$T^{2} +$$$$13\!\cdots\!72$$$$T^{3} +$$$$16\!\cdots\!92$$$$T^{4} +$$$$29\!\cdots\!76$$$$T^{5} +$$$$28\!\cdots\!44$$$$T^{6} +$$$$75\!\cdots\!52$$$$T^{7} + 30136282404129217816 T^{8} - 411910984078744 T^{9} + 119500824744716 T^{10} + 9361537920 T^{11} + 116162 T^{12} - 482 T^{13} + T^{14}$$
$59$ $$35\!\cdots\!72$$$$-$$$$10\!\cdots\!16$$$$T +$$$$15\!\cdots\!24$$$$T^{2} -$$$$13\!\cdots\!56$$$$T^{3} +$$$$80\!\cdots\!12$$$$T^{4} -$$$$31\!\cdots\!96$$$$T^{5} +$$$$82\!\cdots\!28$$$$T^{6} -$$$$18\!\cdots\!16$$$$T^{7} +$$$$70\!\cdots\!88$$$$T^{8} - 2819291680457879576 T^{9} + 687445565640076 T^{10} - 61476408000 T^{11} + 3880898 T^{12} - 2786 T^{13} + T^{14}$$
$61$ $$12\!\cdots\!92$$$$+$$$$21\!\cdots\!00$$$$T +$$$$19\!\cdots\!00$$$$T^{2} -$$$$29\!\cdots\!64$$$$T^{3} +$$$$57\!\cdots\!88$$$$T^{4} +$$$$39\!\cdots\!88$$$$T^{5} +$$$$13\!\cdots\!44$$$$T^{6} -$$$$24\!\cdots\!80$$$$T^{7} +$$$$39\!\cdots\!76$$$$T^{8} + 2978236711893026328 T^{9} + 1123719102824460 T^{10} - 38724599168 T^{11} + 7136642 T^{12} + 3778 T^{13} + T^{14}$$
$67$ $$40\!\cdots\!08$$$$+$$$$14\!\cdots\!84$$$$T +$$$$26\!\cdots\!16$$$$T^{2} +$$$$13\!\cdots\!68$$$$T^{3} +$$$$29\!\cdots\!56$$$$T^{4} -$$$$34\!\cdots\!52$$$$T^{5} +$$$$31\!\cdots\!56$$$$T^{6} +$$$$15\!\cdots\!00$$$$T^{7} +$$$$38\!\cdots\!48$$$$T^{8} - 8669080375183272552 T^{9} + 1177758166491660 T^{10} + 113497816512 T^{11} + 31984002 T^{12} - 7998 T^{13} + T^{14}$$
$71$ $$($$$$38\!\cdots\!80$$$$-$$$$73\!\cdots\!68$$$$T + 3205242620439887904 T^{2} + 628961169111024 T^{3} - 356646354056 T^{4} - 32864444 T^{5} + 9982 T^{6} + T^{7} )^{2}$$
$73$ $$67\!\cdots\!16$$$$+$$$$23\!\cdots\!48$$$$T^{2} +$$$$23\!\cdots\!60$$$$T^{4} +$$$$59\!\cdots\!72$$$$T^{6} +$$$$55\!\cdots\!72$$$$T^{8} + 18004735458743808 T^{10} + 229001536 T^{12} + T^{14}$$
$79$ $$80\!\cdots\!76$$$$+$$$$17\!\cdots\!96$$$$T^{2} +$$$$52\!\cdots\!92$$$$T^{4} +$$$$64\!\cdots\!36$$$$T^{6} +$$$$39\!\cdots\!92$$$$T^{8} + 12122330541457408 T^{10} + 181267456 T^{12} + T^{14}$$
$83$ $$30\!\cdots\!48$$$$-$$$$32\!\cdots\!96$$$$T +$$$$17\!\cdots\!96$$$$T^{2} -$$$$46\!\cdots\!44$$$$T^{3} +$$$$61\!\cdots\!28$$$$T^{4} -$$$$60\!\cdots\!12$$$$T^{5} +$$$$87\!\cdots\!84$$$$T^{6} -$$$$41\!\cdots\!36$$$$T^{7} +$$$$88\!\cdots\!24$$$$T^{8} + 461127974893871208 T^{9} + 129090565299084 T^{10} - 268544938176 T^{11} + 149333762 T^{12} - 17282 T^{13} + T^{14}$$
$89$ $$10\!\cdots\!96$$$$+$$$$14\!\cdots\!80$$$$T^{2} +$$$$54\!\cdots\!80$$$$T^{4} +$$$$52\!\cdots\!20$$$$T^{6} +$$$$20\!\cdots\!84$$$$T^{8} + 39327084461872640 T^{10} + 329862464 T^{12} + T^{14}$$
$97$ $$( -$$$$50\!\cdots\!96$$$$-$$$$13\!\cdots\!60$$$$T - 15741851025018318752 T^{2} + 13158120305248304 T^{3} + 250697829864 T^{4} - 231854348 T^{5} + 2 T^{6} + T^{7} )^{2}$$