# Properties

 Label 16.5.f.a Level $16$ Weight $5$ Character orbit 16.f Analytic conductor $1.654$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 16.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.65391940934$$ 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{21}$$ 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 $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} -\beta_{6} q^{3} + ( -1 + 2 \beta_{3} - \beta_{11} ) q^{4} + ( \beta_{2} + \beta_{3} - \beta_{7} ) q^{5} + ( 5 - \beta_{2} - \beta_{3} - \beta_{10} + \beta_{12} ) q^{6} + ( -1 - 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{7} + ( -6 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{8} + ( \beta_{1} + \beta_{2} - 17 \beta_{3} - 7 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + 2 \beta_{13} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{2} -\beta_{6} q^{3} + ( -1 + 2 \beta_{3} - \beta_{11} ) q^{4} + ( \beta_{2} + \beta_{3} - \beta_{7} ) q^{5} + ( 5 - \beta_{2} - \beta_{3} - \beta_{10} + \beta_{12} ) q^{6} + ( -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} + ( \beta_{1} + \beta_{2} - 17 \beta_{3} - 7 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + 2 \beta_{13} ) q^{9} + ( -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} + ( -23 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{5} + 4 \beta_{6} - \beta_{7} - 4 \beta_{9} - 10 \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{12} + ( -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} + ( 10 + 27 \beta_{1} - 4 \beta_{2} + 21 \beta_{3} + 20 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{10} - 7 \beta_{11} - \beta_{12} - 3 \beta_{13} ) q^{15} + ( -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} + ( 104 - 13 \beta_{1} + 8 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 22 \beta_{6} - 10 \beta_{8} + 2 \beta_{9} - 4 \beta_{10} + 4 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} ) q^{18} + ( -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} + ( 18 + 76 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 30 \beta_{4} + 2 \beta_{5} + 10 \beta_{6} - 4 \beta_{7} + 12 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} + 2 \beta_{12} + 6 \beta_{13} ) q^{21} + ( 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} + ( -140 - 10 \beta_{1} + 8 \beta_{2} + 244 \beta_{3} - 32 \beta_{4} - 14 \beta_{5} + 26 \beta_{6} - 4 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} - 6 \beta_{11} - 6 \beta_{13} ) q^{24} + ( 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} + ( -123 + 49 \beta_{1} + 6 \beta_{2} - 73 \beta_{3} - 113 \beta_{4} - 19 \beta_{5} + 6 \beta_{6} + 12 \beta_{7} + 7 \beta_{8} + 14 \beta_{9} + 4 \beta_{10} - 10 \beta_{11} - 4 \beta_{12} + 6 \beta_{13} ) q^{27} + ( -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} + ( -268 + 36 \beta_{1} - 9 \beta_{2} + 454 \beta_{3} + 8 \beta_{4} + 6 \beta_{5} + 13 \beta_{6} + 32 \beta_{7} + 26 \beta_{8} - 9 \beta_{9} + 3 \beta_{10} - 4 \beta_{11} - 11 \beta_{12} + 3 \beta_{13} ) q^{30} + ( 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} + ( 14 - 69 \beta_{1} - 7 \beta_{2} - 52 \beta_{3} + 125 \beta_{4} + 3 \beta_{5} - 38 \beta_{6} + 11 \beta_{7} - 19 \beta_{8} - \beta_{9} - 32 \beta_{10} + 23 \beta_{11} - 10 \beta_{12} - 16 \beta_{13} ) q^{33} + ( 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} + ( 846 + 4 \beta_{1} - 12 \beta_{2} + 317 \beta_{3} + 104 \beta_{4} + 20 \beta_{5} + 16 \beta_{6} - 5 \beta_{7} - 16 \beta_{8} - 20 \beta_{10} + 28 \beta_{12} - 8 \beta_{13} ) q^{36} + ( -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} + ( 91 - 130 \beta_{1} + 40 \beta_{2} + 32 \beta_{3} - 35 \beta_{4} + 13 \beta_{5} + 20 \beta_{6} - 19 \beta_{7} - 8 \beta_{8} + 24 \beta_{9} + 4 \beta_{10} - 51 \beta_{11} - 5 \beta_{12} + 11 \beta_{13} ) q^{39} + ( -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} + ( -1220 - 6 \beta_{1} + 4 \beta_{2} - 452 \beta_{3} - 2 \beta_{4} - 40 \beta_{6} + 64 \beta_{7} - 8 \beta_{8} - 8 \beta_{9} - 60 \beta_{10} - 32 \beta_{11} - 20 \beta_{12} - 8 \beta_{13} ) q^{42} + ( 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} + ( 111 - 34 \beta_{1} - 18 \beta_{2} + 130 \beta_{3} - 28 \beta_{4} + 4 \beta_{5} - 18 \beta_{6} - 36 \beta_{7} + 2 \beta_{8} - 5 \beta_{9} - 134 \beta_{10} - 17 \beta_{11} + 22 \beta_{12} - 18 \beta_{13} ) q^{45} + ( -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} + ( 528 + 202 \beta_{1} + 26 \beta_{2} - 1452 \beta_{3} - 66 \beta_{4} + 2 \beta_{5} + 86 \beta_{6} + 2 \beta_{7} - 6 \beta_{8} + 30 \beta_{9} + 102 \beta_{10} + 10 \beta_{11} - 14 \beta_{12} + 10 \beta_{13} ) q^{48} + ( 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} + ( -231 + 31 \beta_{1} - 50 \beta_{2} + 217 \beta_{3} - 93 \beta_{4} + 5 \beta_{5} - 10 \beta_{6} + 26 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} + 2 \beta_{12} + 24 \beta_{13} ) q^{51} + ( 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} + ( 824 - 16 \beta_{1} - 34 \beta_{2} - 1920 \beta_{3} - 40 \beta_{4} - 8 \beta_{5} - 98 \beta_{6} + 60 \beta_{7} - 28 \beta_{8} - 14 \beta_{9} + 118 \beta_{10} + 124 \beta_{11} + 2 \beta_{12} - 6 \beta_{13} ) q^{54} + ( -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} + ( -16 - 65 \beta_{1} - \beta_{2} - 98 \beta_{3} + 279 \beta_{4} + 17 \beta_{5} + 176 \beta_{6} + 47 \beta_{7} + 33 \beta_{8} + 31 \beta_{9} - 146 \beta_{10} + 13 \beta_{11} + 16 \beta_{12} + 14 \beta_{13} ) q^{57} + ( -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} + ( -1932 + 482 \beta_{1} + 34 \beta_{2} - 870 \beta_{3} - 322 \beta_{4} - 74 \beta_{5} - 94 \beta_{6} - 36 \beta_{7} + 38 \beta_{8} - 46 \beta_{9} + 46 \beta_{10} + 34 \beta_{11} - 6 \beta_{12} + 46 \beta_{13} ) q^{60} + ( -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} + ( -114 - 317 \beta_{1} + 4 \beta_{2} + 1501 \beta_{3} - 160 \beta_{4} - 46 \beta_{5} - 22 \beta_{6} - 101 \beta_{7} + 47 \beta_{8} - 80 \beta_{9} - 54 \beta_{10} - 51 \beta_{11} - 21 \beta_{12} - 55 \beta_{13} ) q^{63} + ( 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} + ( 2120 - 12 \beta_{1} - 6 \beta_{2} + 1480 \beta_{3} + 70 \beta_{4} + 54 \beta_{5} - 76 \beta_{6} - 92 \beta_{7} + 34 \beta_{8} + 8 \beta_{9} + 78 \beta_{10} - 128 \beta_{11} + 74 \beta_{12} - 52 \beta_{13} ) q^{66} + ( 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} + ( 494 - 604 \beta_{1} + 14 \beta_{2} - 696 \beta_{3} - 278 \beta_{4} - 74 \beta_{5} - 338 \beta_{6} - 112 \beta_{7} - 92 \beta_{8} + 54 \beta_{9} + 54 \beta_{10} + 108 \beta_{11} + 54 \beta_{12} + 98 \beta_{13} ) q^{69} + ( 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} + ( -1118 + 213 \beta_{1} + 75 \beta_{2} + 1886 \beta_{3} + 727 \beta_{4} + 69 \beta_{5} + 205 \beta_{6} + 13 \beta_{7} - 35 \beta_{8} - 51 \beta_{9} - 127 \beta_{10} - 77 \beta_{11} - 37 \beta_{12} + 3 \beta_{13} ) q^{72} + ( -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} + ( 1300 - 52 \beta_{1} - 4 \beta_{2} + 1024 \beta_{3} + 720 \beta_{4} + 112 \beta_{5} - 4 \beta_{6} - 8 \beta_{7} - 12 \beta_{8} - 12 \beta_{9} + 21 \beta_{10} + 128 \beta_{11} - 116 \beta_{12} - 4 \beta_{13} ) q^{75} + ( -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} + ( -558 + 32 \beta_{1} - 119 \beta_{2} + 1164 \beta_{3} + 212 \beta_{4} - 38 \beta_{5} + 105 \beta_{6} + 12 \beta_{7} - 70 \beta_{8} + 35 \beta_{9} + 293 \beta_{10} + 64 \beta_{11} - 21 \beta_{12} - 17 \beta_{13} ) q^{78} + ( -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} + ( -135 + 35 \beta_{1} + 93 \beta_{2} + 280 \beta_{3} - 411 \beta_{4} + 91 \beta_{5} + 298 \beta_{6} - 41 \beta_{7} + 69 \beta_{8} + 83 \beta_{9} + 288 \beta_{10} - 61 \beta_{11} - 42 \beta_{12} - 32 \beta_{13} ) q^{81} + ( 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} + ( 1170 - 360 \beta_{1} - 24 \beta_{2} + 630 \beta_{3} - 1216 \beta_{4} - 120 \beta_{5} + 48 \beta_{6} - 34 \beta_{7} + 32 \beta_{8} - 112 \beta_{9} + 312 \beta_{10} + 86 \beta_{11} + 88 \beta_{12} + 32 \beta_{13} ) q^{84} + ( 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} + ( -3353 + 480 \beta_{1} + 88 \beta_{2} + 94 \beta_{3} - 47 \beta_{4} - 75 \beta_{5} - 62 \beta_{6} + 7 \beta_{7} + 58 \beta_{8} + 28 \beta_{9} - 122 \beta_{10} - 113 \beta_{11} - 35 \beta_{12} + 25 \beta_{13} ) q^{87} + ( -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} + ( -346 + 37 \beta_{1} + \beta_{2} + 218 \beta_{3} + 109 \beta_{4} + 98 \beta_{5} + 405 \beta_{6} - 6 \beta_{7} + 64 \beta_{8} - 137 \beta_{9} - 199 \beta_{10} + 26 \beta_{11} - 53 \beta_{12} - 121 \beta_{13} ) q^{90} + ( -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} + ( 796 + 268 \beta_{1} - 100 \beta_{2} + 716 \beta_{3} - 136 \beta_{4} - 40 \beta_{5} - 100 \beta_{6} - 200 \beta_{7} - 76 \beta_{8} - 32 \beta_{9} + 308 \beta_{10} - 108 \beta_{11} + 140 \beta_{12} - 100 \beta_{13} ) q^{93} + ( 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} + ( -116 - 1396 \beta_{1} - 396 \beta_{3} + 680 \beta_{4} + 28 \beta_{5} - 348 \beta_{6} + 252 \beta_{7} - 88 \beta_{8} + 156 \beta_{9} + 344 \beta_{10} + 64 \beta_{11} - 32 \beta_{12} + 20 \beta_{13} ) q^{96} + ( 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} + ( 3370 - 106 \beta_{1} - 88 \beta_{2} - 3506 \beta_{3} - 334 \beta_{4} - 98 \beta_{5} - 5 \beta_{6} - 156 \beta_{7} + 2 \beta_{8} + 32 \beta_{9} + 32 \beta_{10} + 64 \beta_{11} + 32 \beta_{12} + 244 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q - 2q^{2} - 2q^{3} - 8q^{4} - 2q^{5} + 64q^{6} - 4q^{7} - 92q^{8} + O(q^{10})$$ $$14q - 2q^{2} - 2q^{3} - 8q^{4} - 2q^{5} + 64q^{6} - 4q^{7} - 92q^{8} - 100q^{10} + 94q^{11} - 332q^{12} - 2q^{13} + 44q^{14} - 168q^{16} - 4q^{17} + 1390q^{18} - 706q^{19} + 1900q^{20} - 164q^{21} + 900q^{22} + 1148q^{23} - 1872q^{24} - 3416q^{26} - 1664q^{27} - 3784q^{28} + 862q^{29} - 3740q^{30} + 3208q^{32} - 4q^{33} + 7508q^{34} + 1340q^{35} + 11468q^{36} - 1826q^{37} + 3568q^{38} + 2684q^{39} - 5144q^{40} - 17064q^{42} + 1694q^{43} - 14636q^{44} + 1410q^{45} - 5316q^{46} + 6888q^{48} + 682q^{49} + 20070q^{50} - 3012q^{51} + 20452q^{52} - 482q^{53} + 10784q^{54} - 11780q^{55} - 6952q^{56} - 20456q^{58} - 2786q^{59} - 29920q^{60} - 3778q^{61} - 11472q^{62} + 15808q^{64} - 2020q^{65} + 30148q^{66} + 7998q^{67} + 18032q^{68} + 9628q^{69} + 15296q^{70} + 19964q^{71} - 17708q^{72} - 23780q^{74} + 17570q^{75} - 23996q^{76} - 9508q^{77} - 8052q^{78} + 1384q^{80} + 1454q^{81} + 16016q^{82} - 17282q^{83} + 19624q^{84} + 9948q^{85} - 4796q^{86} - 49284q^{87} + 7288q^{88} - 5416q^{90} - 28036q^{91} - 14632q^{92} + 8896q^{93} + 432q^{94} + 6064q^{96} - 4q^{97} - 12246q^{98} + 49214q^{99} + O(q^{100})$$

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 $$\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$15$$ $$\chi(n)$$ $$-\beta_{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 2.24452 − 1.72109i 0.153862 − 2.82424i 2.79265 + 0.448449i −2.15805 − 1.82834i 1.03712 + 2.63142i 0.336831 + 2.80830i −2.40693 + 1.48549i 2.24452 + 1.72109i 0.153862 + 2.82424i 2.79265 − 0.448449i −2.15805 + 1.82834i 1.03712 − 2.63142i 0.336831 − 2.80830i −2.40693 − 1.48549i
−3.96560 + 0.523430i 5.54016 5.54016i 15.4520 4.15143i 21.7374 21.7374i −19.0702 + 24.8700i −6.62054 −59.1037 + 24.5510i 19.6133i −74.8239 + 97.5799i
3.2 −2.97810 2.67038i −9.42589 + 9.42589i 1.73818 + 15.9053i −2.84710 + 2.84710i 53.2419 2.90058i −76.7794 37.2967 52.0092i 96.6949i 16.0818 0.876123i
3.3 −2.34420 + 3.24110i −4.63552 + 4.63552i −5.00945 15.1956i −29.2002 + 29.2002i −4.15759 25.8908i 59.6196 60.9935 + 19.3854i 38.0239i −26.1896 163.092i
3.4 0.329715 3.98639i 3.91498 3.91498i −15.7826 2.62875i 4.72348 4.72348i −14.3158 16.8975i 45.3712 −15.6830 + 62.0487i 50.3458i −17.2722 20.3870i
3.5 1.59430 + 3.66854i 11.5209 11.5209i −10.9164 + 11.6975i −14.6016 + 14.6016i 60.6325 + 23.8971i −24.0210 −60.3169 21.3980i 184.461i −76.8459 30.2872i
3.6 2.47147 + 3.14513i −7.86839 + 7.86839i −3.78368 + 15.5462i 27.2309 27.2309i −44.1936 5.30063i 50.3097 −58.2460 + 26.5217i 42.8233i 152.945 + 18.3444i
3.7 3.89242 0.921438i −0.0461995 + 0.0461995i 14.3019 7.17325i −8.04297 + 8.04297i −0.137258 + 0.222398i −49.8797 49.0594 41.0996i 80.9957i −23.8955 + 38.7177i
11.1 −3.96560 0.523430i 5.54016 + 5.54016i 15.4520 + 4.15143i 21.7374 + 21.7374i −19.0702 24.8700i −6.62054 −59.1037 24.5510i 19.6133i −74.8239 97.5799i
11.2 −2.97810 + 2.67038i −9.42589 9.42589i 1.73818 15.9053i −2.84710 2.84710i 53.2419 + 2.90058i −76.7794 37.2967 + 52.0092i 96.6949i 16.0818 + 0.876123i
11.3 −2.34420 3.24110i −4.63552 4.63552i −5.00945 + 15.1956i −29.2002 29.2002i −4.15759 + 25.8908i 59.6196 60.9935 19.3854i 38.0239i −26.1896 + 163.092i
11.4 0.329715 + 3.98639i 3.91498 + 3.91498i −15.7826 + 2.62875i 4.72348 + 4.72348i −14.3158 + 16.8975i 45.3712 −15.6830 62.0487i 50.3458i −17.2722 + 20.3870i
11.5 1.59430 3.66854i 11.5209 + 11.5209i −10.9164 11.6975i −14.6016 14.6016i 60.6325 23.8971i −24.0210 −60.3169 + 21.3980i 184.461i −76.8459 + 30.2872i
11.6 2.47147 3.14513i −7.86839 7.86839i −3.78368 15.5462i 27.2309 + 27.2309i −44.1936 + 5.30063i 50.3097 −58.2460 26.5217i 42.8233i 152.945 18.3444i
11.7 3.89242 + 0.921438i −0.0461995 0.0461995i 14.3019 + 7.17325i −8.04297 8.04297i −0.137258 0.222398i −49.8797 49.0594 + 41.0996i 80.9957i −23.8955 38.7177i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.7 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 16.5.f.a 14
3.b odd 2 1 144.5.m.a 14
4.b odd 2 1 64.5.f.a 14
8.b even 2 1 128.5.f.b 14
8.d odd 2 1 128.5.f.a 14
12.b even 2 1 576.5.m.a 14
16.e even 4 1 64.5.f.a 14
16.e even 4 1 128.5.f.a 14
16.f odd 4 1 inner 16.5.f.a 14
16.f odd 4 1 128.5.f.b 14
48.i odd 4 1 576.5.m.a 14
48.k even 4 1 144.5.m.a 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.5.f.a 14 1.a even 1 1 trivial
16.5.f.a 14 16.f odd 4 1 inner
64.5.f.a 14 4.b odd 2 1
64.5.f.a 14 16.e even 4 1
128.5.f.a 14 8.d odd 2 1
128.5.f.a 14 16.e even 4 1
128.5.f.b 14 8.b even 2 1
128.5.f.b 14 16.f odd 4 1
144.5.m.a 14 3.b odd 2 1
144.5.m.a 14 48.k even 4 1
576.5.m.a 14 12.b even 2 1
576.5.m.a 14 48.i odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(16, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 6 T^{2} + 40 T^{3} + 120 T^{4} - 352 T^{5} - 2880 T^{6} + 2560 T^{7} - 46080 T^{8} - 90112 T^{9} + 491520 T^{10} + 2621440 T^{11} + 6291456 T^{12} + 33554432 T^{13} + 268435456 T^{14}$$
$3$ $$1 + 2 T + 2 T^{2} + 610 T^{3} - 4613 T^{4} + 31732 T^{5} + 258740 T^{6} + 2114868 T^{7} + 70347753 T^{8} + 127844190 T^{9} + 3102089886 T^{10} + 22201539966 T^{11} + 75486949083 T^{12} + 2442289052376 T^{13} + 20677791784536 T^{14} + 197825413242456 T^{15} + 495269872933563 T^{16} + 11798808601071006 T^{17} + 133534797839563806 T^{18} + 445765127450480190 T^{19} + 19868283272269877193 T^{20} + 48381396305638460148 T^{21} +$$$$47\!\cdots\!40$$$$T^{22} +$$$$47\!\cdots\!72$$$$T^{23} -$$$$56\!\cdots\!13$$$$T^{24} +$$$$60\!\cdots\!10$$$$T^{25} +$$$$15\!\cdots\!22$$$$T^{26} +$$$$12\!\cdots\!82$$$$T^{27} +$$$$52\!\cdots\!21$$$$T^{28}$$
$5$ $$1 + 2 T + 2 T^{2} + 3938 T^{3} - 94565 T^{4} - 9371916 T^{5} - 10800780 T^{6} - 6597908556 T^{7} - 151886931511 T^{8} + 563896366366 T^{9} + 18344031697054 T^{10} - 1149733079121218 T^{11} + 70375486014886875 T^{12} + 1339233603531363800 T^{13} + 14585682673141755608 T^{14} +$$$$83\!\cdots\!00$$$$T^{15} +$$$$27\!\cdots\!75$$$$T^{16} -$$$$28\!\cdots\!50$$$$T^{17} +$$$$27\!\cdots\!50$$$$T^{18} +$$$$53\!\cdots\!50$$$$T^{19} -$$$$90\!\cdots\!75$$$$T^{20} -$$$$24\!\cdots\!00$$$$T^{21} -$$$$25\!\cdots\!00$$$$T^{22} -$$$$13\!\cdots\!00$$$$T^{23} -$$$$86\!\cdots\!25$$$$T^{24} +$$$$22\!\cdots\!50$$$$T^{25} +$$$$71\!\cdots\!50$$$$T^{26} +$$$$44\!\cdots\!50$$$$T^{27} +$$$$13\!\cdots\!25$$$$T^{28}$$
$7$ $$( 1 + 2 T + 8235 T^{2} + 64404 T^{3} + 38860249 T^{4} + 447351454 T^{5} + 125587217723 T^{6} + 1378109878936 T^{7} + 301534909752923 T^{8} + 2578892109370654 T^{9} + 537875867111373049 T^{10} + 2140333660404582804 T^{11} +$$$$65\!\cdots\!35$$$$T^{12} +$$$$38\!\cdots\!02$$$$T^{13} +$$$$45\!\cdots\!01$$$$T^{14} )^{2}$$
$11$ $$1 - 94 T + 4418 T^{2} + 965570 T^{3} - 470088133 T^{4} + 3120961844 T^{5} + 2249641670708 T^{6} - 796542815073292 T^{7} + 97369777194709097 T^{8} + 5317394377195027646 T^{9} -$$$$63\!\cdots\!78$$$$T^{10} +$$$$14\!\cdots\!86$$$$T^{11} -$$$$23\!\cdots\!21$$$$T^{12} -$$$$22\!\cdots\!52$$$$T^{13} +$$$$18\!\cdots\!92$$$$T^{14} -$$$$33\!\cdots\!32$$$$T^{15} -$$$$50\!\cdots\!01$$$$T^{16} +$$$$45\!\cdots\!06$$$$T^{17} -$$$$29\!\cdots\!58$$$$T^{18} +$$$$35\!\cdots\!46$$$$T^{19} +$$$$95\!\cdots\!77$$$$T^{20} -$$$$11\!\cdots\!52$$$$T^{21} +$$$$47\!\cdots\!68$$$$T^{22} +$$$$96\!\cdots\!84$$$$T^{23} -$$$$21\!\cdots\!33$$$$T^{24} +$$$$63\!\cdots\!70$$$$T^{25} +$$$$42\!\cdots\!58$$$$T^{26} -$$$$13\!\cdots\!74$$$$T^{27} +$$$$20\!\cdots\!61$$$$T^{28}$$
$13$ $$1 + 2 T + 2 T^{2} + 6883234 T^{3} + 1464853339 T^{4} + 65707775476 T^{5} + 23817940993652 T^{6} + 16199073445624116 T^{7} + 1142495439970904649 T^{8} +$$$$10\!\cdots\!70$$$$T^{9} +$$$$78\!\cdots\!46$$$$T^{10} +$$$$14\!\cdots\!90$$$$T^{11} +$$$$74\!\cdots\!35$$$$T^{12} +$$$$22\!\cdots\!32$$$$T^{13} +$$$$94\!\cdots\!60$$$$T^{14} +$$$$65\!\cdots\!52$$$$T^{15} +$$$$60\!\cdots\!35$$$$T^{16} +$$$$33\!\cdots\!90$$$$T^{17} +$$$$52\!\cdots\!86$$$$T^{18} +$$$$19\!\cdots\!70$$$$T^{19} +$$$$62\!\cdots\!89$$$$T^{20} +$$$$25\!\cdots\!36$$$$T^{21} +$$$$10\!\cdots\!12$$$$T^{22} +$$$$83\!\cdots\!16$$$$T^{23} +$$$$52\!\cdots\!39$$$$T^{24} +$$$$71\!\cdots\!74$$$$T^{25} +$$$$58\!\cdots\!42$$$$T^{26} +$$$$16\!\cdots\!62$$$$T^{27} +$$$$24\!\cdots\!41$$$$T^{28}$$
$17$ $$( 1 + 2 T + 333755 T^{2} - 12776716 T^{3} + 56419031945 T^{4} - 2961382342882 T^{5} + 6454907058757691 T^{6} - 326099715157486120 T^{7} +$$$$53\!\cdots\!11$$$$T^{8} -$$$$20\!\cdots\!62$$$$T^{9} +$$$$32\!\cdots\!45$$$$T^{10} -$$$$62\!\cdots\!96$$$$T^{11} +$$$$13\!\cdots\!55$$$$T^{12} +$$$$67\!\cdots\!42$$$$T^{13} +$$$$28\!\cdots\!41$$$$T^{14} )^{2}$$
$19$ $$1 + 706 T + 249218 T^{2} + 101381538 T^{3} + 32599242619 T^{4} + 3617133340788 T^{5} - 431513784802892 T^{6} - 960761925731564364 T^{7} -$$$$71\!\cdots\!35$$$$T^{8} -$$$$22\!\cdots\!14$$$$T^{9} -$$$$57\!\cdots\!66$$$$T^{10} -$$$$18\!\cdots\!22$$$$T^{11} -$$$$19\!\cdots\!61$$$$T^{12} +$$$$38\!\cdots\!56$$$$T^{13} +$$$$11\!\cdots\!48$$$$T^{14} +$$$$50\!\cdots\!76$$$$T^{15} -$$$$32\!\cdots\!01$$$$T^{16} -$$$$40\!\cdots\!42$$$$T^{17} -$$$$16\!\cdots\!46$$$$T^{18} -$$$$84\!\cdots\!14$$$$T^{19} -$$$$35\!\cdots\!35$$$$T^{20} -$$$$61\!\cdots\!24$$$$T^{21} -$$$$35\!\cdots\!12$$$$T^{22} +$$$$39\!\cdots\!28$$$$T^{23} +$$$$46\!\cdots\!19$$$$T^{24} +$$$$18\!\cdots\!98$$$$T^{25} +$$$$59\!\cdots\!38$$$$T^{26} +$$$$22\!\cdots\!66$$$$T^{27} +$$$$40\!\cdots\!81$$$$T^{28}$$
$23$ $$( 1 - 574 T + 1024043 T^{2} - 635922028 T^{3} + 551773439769 T^{4} - 314763003369506 T^{5} + 213700110666561659 T^{6} -$$$$10\!\cdots\!08$$$$T^{7} +$$$$59\!\cdots\!19$$$$T^{8} -$$$$24\!\cdots\!86$$$$T^{9} +$$$$12\!\cdots\!49$$$$T^{10} -$$$$38\!\cdots\!08$$$$T^{11} +$$$$17\!\cdots\!43$$$$T^{12} -$$$$27\!\cdots\!34$$$$T^{13} +$$$$13\!\cdots\!81$$$$T^{14} )^{2}$$
$29$ $$1 - 862 T + 371522 T^{2} - 1045006654 T^{3} + 1721779716827 T^{4} - 691721668187596 T^{5} + 502604487474844916 T^{6} -$$$$98\!\cdots\!28$$$$T^{7} +$$$$75\!\cdots\!49$$$$T^{8} -$$$$31\!\cdots\!74$$$$T^{9} +$$$$39\!\cdots\!98$$$$T^{10} -$$$$45\!\cdots\!94$$$$T^{11} +$$$$34\!\cdots\!87$$$$T^{12} -$$$$29\!\cdots\!64$$$$T^{13} +$$$$24\!\cdots\!96$$$$T^{14} -$$$$20\!\cdots\!84$$$$T^{15} +$$$$17\!\cdots\!07$$$$T^{16} -$$$$15\!\cdots\!54$$$$T^{17} +$$$$98\!\cdots\!58$$$$T^{18} -$$$$54\!\cdots\!74$$$$T^{19} +$$$$94\!\cdots\!69$$$$T^{20} -$$$$87\!\cdots\!08$$$$T^{21} +$$$$31\!\cdots\!56$$$$T^{22} -$$$$30\!\cdots\!16$$$$T^{23} +$$$$53\!\cdots\!27$$$$T^{24} -$$$$23\!\cdots\!74$$$$T^{25} +$$$$58\!\cdots\!42$$$$T^{26} -$$$$95\!\cdots\!42$$$$T^{27} +$$$$78\!\cdots\!21$$$$T^{28}$$
$31$ $$1 - 6904334 T^{2} + 24182883262811 T^{4} - 57459081771770667372 T^{6} +$$$$10\!\cdots\!69$$$$T^{8} -$$$$14\!\cdots\!50$$$$T^{10} +$$$$17\!\cdots\!83$$$$T^{12} -$$$$17\!\cdots\!20$$$$T^{14} +$$$$15\!\cdots\!03$$$$T^{16} -$$$$10\!\cdots\!50$$$$T^{18} +$$$$64\!\cdots\!49$$$$T^{20} -$$$$30\!\cdots\!92$$$$T^{22} +$$$$10\!\cdots\!11$$$$T^{24} -$$$$26\!\cdots\!94$$$$T^{26} +$$$$32\!\cdots\!81$$$$T^{28}$$
$37$ $$1 + 1826 T + 1667138 T^{2} + 4976934274 T^{3} + 7030206539163 T^{4} + 1716272691299380 T^{5} + 3798526848883495924 T^{6} +$$$$11\!\cdots\!60$$$$T^{7} -$$$$10\!\cdots\!63$$$$T^{8} -$$$$16\!\cdots\!22$$$$T^{9} +$$$$18\!\cdots\!50$$$$T^{10} -$$$$21\!\cdots\!18$$$$T^{11} -$$$$10\!\cdots\!97$$$$T^{12} -$$$$14\!\cdots\!60$$$$T^{13} -$$$$67\!\cdots\!04$$$$T^{14} -$$$$26\!\cdots\!60$$$$T^{15} -$$$$37\!\cdots\!37$$$$T^{16} -$$$$14\!\cdots\!58$$$$T^{17} +$$$$22\!\cdots\!50$$$$T^{18} -$$$$38\!\cdots\!22$$$$T^{19} -$$$$44\!\cdots\!43$$$$T^{20} +$$$$91\!\cdots\!60$$$$T^{21} +$$$$57\!\cdots\!44$$$$T^{22} +$$$$48\!\cdots\!80$$$$T^{23} +$$$$37\!\cdots\!63$$$$T^{24} +$$$$49\!\cdots\!14$$$$T^{25} +$$$$31\!\cdots\!98$$$$T^{26} +$$$$64\!\cdots\!06$$$$T^{27} +$$$$65\!\cdots\!41$$$$T^{28}$$
$41$ $$1 - 24523982 T^{2} + 302442312166171 T^{4} -$$$$24\!\cdots\!76$$$$T^{6} +$$$$14\!\cdots\!01$$$$T^{8} -$$$$70\!\cdots\!70$$$$T^{10} +$$$$27\!\cdots\!51$$$$T^{12} -$$$$84\!\cdots\!88$$$$T^{14} +$$$$21\!\cdots\!71$$$$T^{16} -$$$$45\!\cdots\!70$$$$T^{18} +$$$$76\!\cdots\!61$$$$T^{20} -$$$$10\!\cdots\!56$$$$T^{22} +$$$$98\!\cdots\!71$$$$T^{24} -$$$$63\!\cdots\!22$$$$T^{26} +$$$$20\!\cdots\!41$$$$T^{28}$$
$43$ $$1 - 1694 T + 1434818 T^{2} - 14278395262 T^{3} + 44454402050619 T^{4} - 13474016894363980 T^{5} + 60977294006539554100 T^{6} -$$$$38\!\cdots\!44$$$$T^{7} +$$$$23\!\cdots\!81$$$$T^{8} +$$$$78\!\cdots\!82$$$$T^{9} +$$$$12\!\cdots\!62$$$$T^{10} -$$$$18\!\cdots\!22$$$$T^{11} -$$$$91\!\cdots\!45$$$$T^{12} +$$$$10\!\cdots\!08$$$$T^{13} +$$$$16\!\cdots\!52$$$$T^{14} +$$$$36\!\cdots\!08$$$$T^{15} -$$$$10\!\cdots\!45$$$$T^{16} -$$$$73\!\cdots\!22$$$$T^{17} +$$$$16\!\cdots\!62$$$$T^{18} +$$$$36\!\cdots\!82$$$$T^{19} +$$$$37\!\cdots\!81$$$$T^{20} -$$$$21\!\cdots\!44$$$$T^{21} +$$$$11\!\cdots\!00$$$$T^{22} -$$$$85\!\cdots\!80$$$$T^{23} +$$$$96\!\cdots\!19$$$$T^{24} -$$$$10\!\cdots\!62$$$$T^{25} +$$$$36\!\cdots\!18$$$$T^{26} -$$$$14\!\cdots\!94$$$$T^{27} +$$$$29\!\cdots\!01$$$$T^{28}$$
$47$ $$1 - 51887758 T^{2} + 1309844227745755 T^{4} -$$$$21\!\cdots\!56$$$$T^{6} +$$$$24\!\cdots\!09$$$$T^{8} -$$$$21\!\cdots\!30$$$$T^{10} +$$$$15\!\cdots\!39$$$$T^{12} -$$$$82\!\cdots\!04$$$$T^{14} +$$$$35\!\cdots\!79$$$$T^{16} -$$$$12\!\cdots\!30$$$$T^{18} +$$$$33\!\cdots\!29$$$$T^{20} -$$$$68\!\cdots\!96$$$$T^{22} +$$$$10\!\cdots\!55$$$$T^{24} -$$$$94\!\cdots\!38$$$$T^{26} +$$$$43\!\cdots\!21$$$$T^{28}$$
$53$ $$1 + 482 T + 116162 T^{2} - 5558326078 T^{3} + 43583027341595 T^{4} + 155411473123116980 T^{5} + 85293132817975457012 T^{6} +$$$$11\!\cdots\!76$$$$T^{7} +$$$$35\!\cdots\!13$$$$T^{8} -$$$$92\!\cdots\!94$$$$T^{9} +$$$$12\!\cdots\!18$$$$T^{10} +$$$$13\!\cdots\!10$$$$T^{11} +$$$$28\!\cdots\!31$$$$T^{12} +$$$$18\!\cdots\!24$$$$T^{13} +$$$$12\!\cdots\!68$$$$T^{14} +$$$$14\!\cdots\!44$$$$T^{15} +$$$$17\!\cdots\!91$$$$T^{16} +$$$$67\!\cdots\!10$$$$T^{17} +$$$$47\!\cdots\!78$$$$T^{18} -$$$$28\!\cdots\!94$$$$T^{19} +$$$$84\!\cdots\!53$$$$T^{20} +$$$$21\!\cdots\!36$$$$T^{21} +$$$$12\!\cdots\!92$$$$T^{22} +$$$$18\!\cdots\!80$$$$T^{23} +$$$$40\!\cdots\!95$$$$T^{24} -$$$$41\!\cdots\!18$$$$T^{25} +$$$$67\!\cdots\!82$$$$T^{26} +$$$$22\!\cdots\!62$$$$T^{27} +$$$$36\!\cdots\!21$$$$T^{28}$$
$59$ $$1 + 2786 T + 3880898 T^{2} + 95235375746 T^{3} + 282835430943931 T^{4} - 951817240129082700 T^{5} +$$$$78\!\cdots\!20$$$$T^{6} -$$$$10\!\cdots\!24$$$$T^{7} -$$$$10\!\cdots\!59$$$$T^{8} -$$$$16\!\cdots\!78$$$$T^{9} +$$$$95\!\cdots\!50$$$$T^{10} -$$$$27\!\cdots\!94$$$$T^{11} +$$$$66\!\cdots\!23$$$$T^{12} +$$$$40\!\cdots\!76$$$$T^{13} +$$$$34\!\cdots\!76$$$$T^{14} +$$$$49\!\cdots\!36$$$$T^{15} +$$$$97\!\cdots\!83$$$$T^{16} -$$$$49\!\cdots\!14$$$$T^{17} +$$$$20\!\cdots\!50$$$$T^{18} -$$$$42\!\cdots\!78$$$$T^{19} -$$$$34\!\cdots\!99$$$$T^{20} -$$$$39\!\cdots\!04$$$$T^{21} +$$$$36\!\cdots\!20$$$$T^{22} -$$$$53\!\cdots\!00$$$$T^{23} +$$$$19\!\cdots\!31$$$$T^{24} +$$$$78\!\cdots\!06$$$$T^{25} +$$$$38\!\cdots\!58$$$$T^{26} +$$$$33\!\cdots\!66$$$$T^{27} +$$$$14\!\cdots\!41$$$$T^{28}$$
$61$ $$1 + 3778 T + 7136642 T^{2} + 13584988130 T^{3} + 15885658886619 T^{4} - 322904339201283724 T^{5} -$$$$12\!\cdots\!20$$$$T^{6} -$$$$11\!\cdots\!80$$$$T^{7} -$$$$34\!\cdots\!59$$$$T^{8} -$$$$57\!\cdots\!82$$$$T^{9} -$$$$60\!\cdots\!50$$$$T^{10} -$$$$13\!\cdots\!70$$$$T^{11} +$$$$15\!\cdots\!55$$$$T^{12} +$$$$19\!\cdots\!60$$$$T^{13} +$$$$92\!\cdots\!08$$$$T^{14} +$$$$26\!\cdots\!60$$$$T^{15} +$$$$29\!\cdots\!55$$$$T^{16} -$$$$35\!\cdots\!70$$$$T^{17} -$$$$22\!\cdots\!50$$$$T^{18} -$$$$29\!\cdots\!82$$$$T^{19} -$$$$24\!\cdots\!19$$$$T^{20} -$$$$11\!\cdots\!80$$$$T^{21} -$$$$16\!\cdots\!20$$$$T^{22} -$$$$60\!\cdots\!64$$$$T^{23} +$$$$41\!\cdots\!19$$$$T^{24} +$$$$48\!\cdots\!30$$$$T^{25} +$$$$35\!\cdots\!02$$$$T^{26} +$$$$25\!\cdots\!38$$$$T^{27} +$$$$95\!\cdots\!61$$$$T^{28}$$
$67$ $$1 - 7998 T + 31984002 T^{2} - 47670849246 T^{3} - 439076236005637 T^{4} + 648765759780793716 T^{5} +$$$$99\!\cdots\!64$$$$T^{6} -$$$$93\!\cdots\!52$$$$T^{7} +$$$$28\!\cdots\!21$$$$T^{8} +$$$$69\!\cdots\!70$$$$T^{9} -$$$$55\!\cdots\!14$$$$T^{10} +$$$$21\!\cdots\!74$$$$T^{11} -$$$$39\!\cdots\!49$$$$T^{12} +$$$$17\!\cdots\!60$$$$T^{13} -$$$$39\!\cdots\!76$$$$T^{14} +$$$$35\!\cdots\!60$$$$T^{15} -$$$$15\!\cdots\!09$$$$T^{16} +$$$$17\!\cdots\!14$$$$T^{17} -$$$$92\!\cdots\!34$$$$T^{18} +$$$$23\!\cdots\!70$$$$T^{19} +$$$$18\!\cdots\!41$$$$T^{20} -$$$$12\!\cdots\!32$$$$T^{21} +$$$$27\!\cdots\!04$$$$T^{22} +$$$$35\!\cdots\!96$$$$T^{23} -$$$$48\!\cdots\!37$$$$T^{24} -$$$$10\!\cdots\!66$$$$T^{25} +$$$$14\!\cdots\!82$$$$T^{26} -$$$$72\!\cdots\!78$$$$T^{27} +$$$$18\!\cdots\!81$$$$T^{28}$$
$71$ $$( 1 - 9982 T + 145017323 T^{2} - 1165310044396 T^{3} + 10014081489420185 T^{4} - 63641985337531169890 T^{5} +$$$$40\!\cdots\!59$$$$T^{6} -$$$$20\!\cdots\!72$$$$T^{7} +$$$$10\!\cdots\!79$$$$T^{8} -$$$$41\!\cdots\!90$$$$T^{9} +$$$$16\!\cdots\!85$$$$T^{10} -$$$$48\!\cdots\!16$$$$T^{11} +$$$$15\!\cdots\!23$$$$T^{12} -$$$$26\!\cdots\!42$$$$T^{13} +$$$$68\!\cdots\!61$$$$T^{14} )^{2}$$
$73$ $$1 - 168573838 T^{2} + 13353714116727067 T^{4} -$$$$70\!\cdots\!96$$$$T^{6} +$$$$30\!\cdots\!57$$$$T^{8} -$$$$11\!\cdots\!98$$$$T^{10} +$$$$39\!\cdots\!07$$$$T^{12} -$$$$11\!\cdots\!44$$$$T^{14} +$$$$31\!\cdots\!67$$$$T^{16} -$$$$74\!\cdots\!78$$$$T^{18} +$$$$15\!\cdots\!37$$$$T^{20} -$$$$29\!\cdots\!16$$$$T^{22} +$$$$45\!\cdots\!67$$$$T^{24} -$$$$46\!\cdots\!78$$$$T^{26} +$$$$22\!\cdots\!61$$$$T^{28}$$
$79$ $$1 - 364033678 T^{2} + 65454647116587227 T^{4} -$$$$76\!\cdots\!56$$$$T^{6} +$$$$65\!\cdots\!81$$$$T^{8} -$$$$43\!\cdots\!10$$$$T^{10} +$$$$23\!\cdots\!67$$$$T^{12} -$$$$99\!\cdots\!88$$$$T^{14} +$$$$35\!\cdots\!87$$$$T^{16} -$$$$10\!\cdots\!10$$$$T^{18} +$$$$23\!\cdots\!61$$$$T^{20} -$$$$40\!\cdots\!96$$$$T^{22} +$$$$52\!\cdots\!27$$$$T^{24} -$$$$44\!\cdots\!58$$$$T^{26} +$$$$18\!\cdots\!21$$$$T^{28}$$
$83$ $$1 + 17282 T + 149333762 T^{2} + 1088719641698 T^{3} + 16964332297412731 T^{4} +$$$$21\!\cdots\!96$$$$T^{5} +$$$$17\!\cdots\!52$$$$T^{6} +$$$$12\!\cdots\!12$$$$T^{7} +$$$$11\!\cdots\!61$$$$T^{8} +$$$$11\!\cdots\!62$$$$T^{9} +$$$$90\!\cdots\!74$$$$T^{10} +$$$$61\!\cdots\!82$$$$T^{11} +$$$$44\!\cdots\!67$$$$T^{12} +$$$$35\!\cdots\!48$$$$T^{13} +$$$$26\!\cdots\!76$$$$T^{14} +$$$$16\!\cdots\!08$$$$T^{15} +$$$$10\!\cdots\!47$$$$T^{16} +$$$$65\!\cdots\!02$$$$T^{17} +$$$$45\!\cdots\!94$$$$T^{18} +$$$$28\!\cdots\!62$$$$T^{19} +$$$$13\!\cdots\!81$$$$T^{20} +$$$$66\!\cdots\!92$$$$T^{21} +$$$$44\!\cdots\!72$$$$T^{22} +$$$$26\!\cdots\!76$$$$T^{23} +$$$$98\!\cdots\!31$$$$T^{24} +$$$$29\!\cdots\!58$$$$T^{25} +$$$$19\!\cdots\!42$$$$T^{26} +$$$$10\!\cdots\!02$$$$T^{27} +$$$$29\!\cdots\!81$$$$T^{28}$$
$89$ $$1 - 548528910 T^{2} + 149200943223060123 T^{4} -$$$$26\!\cdots\!16$$$$T^{6} +$$$$35\!\cdots\!49$$$$T^{8} -$$$$37\!\cdots\!50$$$$T^{10} +$$$$31\!\cdots\!11$$$$T^{12} -$$$$21\!\cdots\!00$$$$T^{14} +$$$$12\!\cdots\!91$$$$T^{16} -$$$$57\!\cdots\!50$$$$T^{18} +$$$$21\!\cdots\!09$$$$T^{20} -$$$$64\!\cdots\!36$$$$T^{22} +$$$$14\!\cdots\!23$$$$T^{24} -$$$$20\!\cdots\!10$$$$T^{26} +$$$$14\!\cdots\!61$$$$T^{28}$$
$97$ $$( 1 + 2 T + 387850619 T^{2} + 251760181236 T^{3} + 75114732161345545 T^{4} + 73269666487293981214 T^{5} +$$$$94\!\cdots\!67$$$$T^{6} +$$$$89\!\cdots\!44$$$$T^{7} +$$$$83\!\cdots\!27$$$$T^{8} +$$$$57\!\cdots\!54$$$$T^{9} +$$$$52\!\cdots\!45$$$$T^{10} +$$$$15\!\cdots\!56$$$$T^{11} +$$$$21\!\cdots\!19$$$$T^{12} +$$$$96\!\cdots\!62$$$$T^{13} +$$$$42\!\cdots\!61$$$$T^{14} )^{2}$$