# Properties

 Label 60.5.c.a Level $60$ Weight $5$ Character orbit 60.c Analytic conductor $6.202$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 60.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.20219778503$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} - 9 x^{14} + 18 x^{13} + 263 x^{12} - 444 x^{11} - 1732 x^{10} - 832 x^{9} + 25296 x^{8} - 6656 x^{7} - 110848 x^{6} - 227328 x^{5} + 1077248 x^{4} + 589824 x^{3} - 2359296 x^{2} - 8388608 x + 16777216$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{32}\cdot 3^{4}\cdot 5^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{3} ) q^{2} + \beta_{2} q^{3} + ( 2 + \beta_{3} + \beta_{7} ) q^{4} -\beta_{8} q^{5} + ( 1 - \beta_{2} - \beta_{9} ) q^{6} + ( -1 - 3 \beta_{3} - \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{7} + ( -11 + \beta_{1} + 5 \beta_{2} - \beta_{3} - \beta_{8} - \beta_{10} + \beta_{12} ) q^{8} -27 q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{3} ) q^{2} + \beta_{2} q^{3} + ( 2 + \beta_{3} + \beta_{7} ) q^{4} -\beta_{8} q^{5} + ( 1 - \beta_{2} - \beta_{9} ) q^{6} + ( -1 - 3 \beta_{3} - \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{7} + ( -11 + \beta_{1} + 5 \beta_{2} - \beta_{3} - \beta_{8} - \beta_{10} + \beta_{12} ) q^{8} -27 q^{9} + ( 3 - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{10} + ( 1 + \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{6} - 3 \beta_{8} - 2 \beta_{10} - \beta_{11} + \beta_{12} + 4 \beta_{13} + 2 \beta_{14} ) q^{11} + ( \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{12} + ( -22 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} + 5 \beta_{6} - \beta_{7} - 6 \beta_{8} + 4 \beta_{10} - 2 \beta_{11} - \beta_{13} + 5 \beta_{14} + 3 \beta_{15} ) q^{13} + ( -49 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 6 \beta_{8} + \beta_{9} + 2 \beta_{10} - 5 \beta_{11} + 3 \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{14} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} - \beta_{8} - \beta_{13} ) q^{15} + ( -13 - 2 \beta_{1} + 5 \beta_{2} + 8 \beta_{3} + \beta_{4} + 6 \beta_{6} - 3 \beta_{7} - 10 \beta_{8} - 7 \beta_{9} + 3 \beta_{10} - 6 \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{16} + ( -7 + 6 \beta_{1} + \beta_{2} - 20 \beta_{3} - 6 \beta_{5} + 7 \beta_{6} + 6 \beta_{8} + 3 \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} + 6 \beta_{15} ) q^{17} + ( 27 + 27 \beta_{3} ) q^{18} + ( -5 + 4 \beta_{1} + 23 \beta_{2} - 4 \beta_{3} + \beta_{5} - 2 \beta_{6} + 6 \beta_{8} + 3 \beta_{9} + \beta_{10} + 3 \beta_{11} - 10 \beta_{12} + \beta_{13} - \beta_{15} ) q^{19} + ( 37 - 2 \beta_{1} - 6 \beta_{2} - 10 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} - 9 \beta_{8} + \beta_{9} + \beta_{10} - 5 \beta_{11} + 4 \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{20} + ( 16 - 2 \beta_{1} - 3 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 4 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{12} - \beta_{13} + 5 \beta_{14} + 4 \beta_{15} ) q^{21} + ( 22 - 2 \beta_{1} + 14 \beta_{2} - 4 \beta_{3} - 12 \beta_{4} - 6 \beta_{5} + 4 \beta_{6} + 16 \beta_{7} + 20 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 6 \beta_{12} - 6 \beta_{13} - 6 \beta_{14} - 10 \beta_{15} ) q^{22} + ( -3 - 2 \beta_{1} - 18 \beta_{2} - 11 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} - 16 \beta_{7} - 9 \beta_{8} + 10 \beta_{9} - 3 \beta_{10} - \beta_{11} - 4 \beta_{12} + 12 \beta_{13} + 8 \beta_{14} + \beta_{15} ) q^{23} + ( -120 - 6 \beta_{1} - 11 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + 5 \beta_{12} - 6 \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{24} + 125 q^{25} + ( -39 + 4 \beta_{1} + 12 \beta_{2} + 34 \beta_{3} + \beta_{4} - 2 \beta_{5} - 14 \beta_{6} + 4 \beta_{7} + 12 \beta_{8} - \beta_{9} - 14 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} + 9 \beta_{13} - 16 \beta_{14} - 2 \beta_{15} ) q^{26} -27 \beta_{2} q^{27} + ( -54 + 12 \beta_{1} + 70 \beta_{3} + 6 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} - 10 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 6 \beta_{11} - 10 \beta_{13} - 16 \beta_{14} - 8 \beta_{15} ) q^{28} + ( -218 - 16 \beta_{1} - 4 \beta_{2} - 44 \beta_{3} + 20 \beta_{4} + 18 \beta_{5} - 4 \beta_{6} - 22 \beta_{7} - 26 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + 14 \beta_{11} - 6 \beta_{12} + 2 \beta_{13} + 8 \beta_{14} + 14 \beta_{15} ) q^{29} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} + \beta_{14} + 5 \beta_{15} ) q^{30} + ( -20 + 8 \beta_{1} - 47 \beta_{2} - 42 \beta_{3} - 13 \beta_{4} - \beta_{5} - 10 \beta_{6} + 24 \beta_{7} + 32 \beta_{8} - 8 \beta_{9} + 5 \beta_{10} + 16 \beta_{11} - 17 \beta_{12} - 6 \beta_{13} - 6 \beta_{14} - 5 \beta_{15} ) q^{31} + ( 488 + 15 \beta_{1} - 49 \beta_{2} + 11 \beta_{3} - 3 \beta_{4} - 14 \beta_{5} - 26 \beta_{6} + 6 \beta_{7} + 21 \beta_{8} - 3 \beta_{9} - 9 \beta_{10} + 5 \beta_{11} + 14 \beta_{12} - 17 \beta_{13} - 16 \beta_{14} - 6 \beta_{15} ) q^{32} + ( 14 + 12 \beta_{1} - 2 \beta_{2} + 36 \beta_{3} + \beta_{4} - 11 \beta_{5} - 26 \beta_{6} + 9 \beta_{7} + 20 \beta_{8} + 5 \beta_{9} - 15 \beta_{10} + 18 \beta_{11} - 2 \beta_{12} - 10 \beta_{14} + \beta_{15} ) q^{33} + ( 303 - 32 \beta_{1} + 80 \beta_{2} - 22 \beta_{3} - \beta_{4} + 26 \beta_{5} - 22 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + 9 \beta_{9} - 2 \beta_{10} + 5 \beta_{11} + \beta_{12} - \beta_{13} - 4 \beta_{14} - 14 \beta_{15} ) q^{34} + ( -15 - 6 \beta_{1} + 11 \beta_{2} - 70 \beta_{3} - 6 \beta_{4} - 5 \beta_{5} - 10 \beta_{6} + 6 \beta_{7} + 6 \beta_{8} + 15 \beta_{9} - 5 \beta_{10} + 5 \beta_{11} + 10 \beta_{12} - 9 \beta_{13} - 15 \beta_{15} ) q^{35} + ( -54 - 27 \beta_{3} - 27 \beta_{7} ) q^{36} + ( 540 + 6 \beta_{1} + 3 \beta_{2} - 150 \beta_{3} + 8 \beta_{4} - 9 \beta_{5} + 13 \beta_{6} - 27 \beta_{7} - 54 \beta_{8} - 12 \beta_{9} + 2 \beta_{10} - 10 \beta_{12} + 13 \beta_{13} - 7 \beta_{14} + 13 \beta_{15} ) q^{37} + ( -83 - 12 \beta_{1} - 60 \beta_{2} + 26 \beta_{3} - 33 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} - 14 \beta_{8} - 27 \beta_{9} + 20 \beta_{10} - 3 \beta_{11} - 3 \beta_{12} - \beta_{13} + 12 \beta_{14} + 22 \beta_{15} ) q^{38} + ( 6 - 2 \beta_{1} - 23 \beta_{2} - 24 \beta_{3} + 7 \beta_{4} + 15 \beta_{5} - 6 \beta_{6} - 22 \beta_{7} - 16 \beta_{8} + 12 \beta_{9} + 9 \beta_{10} - 18 \beta_{11} + 9 \beta_{12} + 8 \beta_{13} + 6 \beta_{14} + 3 \beta_{15} ) q^{39} + ( 24 + 12 \beta_{1} + 3 \beta_{2} - 30 \beta_{3} + 2 \beta_{4} - 8 \beta_{5} + 21 \beta_{7} + 26 \beta_{8} - 3 \beta_{10} + 5 \beta_{11} - 5 \beta_{12} - 12 \beta_{13} - 15 \beta_{14} ) q^{40} + ( 150 + 20 \beta_{1} + 14 \beta_{2} + 232 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} - 18 \beta_{6} + 22 \beta_{7} + 20 \beta_{8} + 32 \beta_{9} + 4 \beta_{10} + 36 \beta_{11} + 16 \beta_{12} - 2 \beta_{13} - 10 \beta_{14} - 18 \beta_{15} ) q^{41} + ( -13 - 8 \beta_{1} - 32 \beta_{2} - 12 \beta_{3} + \beta_{4} - 6 \beta_{5} - 18 \beta_{6} + 8 \beta_{7} - 16 \beta_{8} + 7 \beta_{9} - 18 \beta_{10} + 3 \beta_{11} + 15 \beta_{12} + 17 \beta_{13} - 12 \beta_{14} - 18 \beta_{15} ) q^{42} + ( 14 + 4 \beta_{1} + 74 \beta_{2} + 166 \beta_{3} - 14 \beta_{4} + 4 \beta_{5} + 20 \beta_{6} - 28 \beta_{7} + 6 \beta_{8} - 12 \beta_{9} + 8 \beta_{10} + 10 \beta_{11} - 46 \beta_{12} + 8 \beta_{13} - 4 \beta_{14} + 28 \beta_{15} ) q^{43} + ( -394 + 8 \beta_{1} - 12 \beta_{2} - 14 \beta_{3} + 16 \beta_{4} + 38 \beta_{5} - 40 \beta_{6} - 26 \beta_{7} + 58 \beta_{8} + 14 \beta_{9} + 10 \beta_{10} - 18 \beta_{11} - 12 \beta_{12} - 26 \beta_{13} - 8 \beta_{14} - 8 \beta_{15} ) q^{44} + 27 \beta_{8} q^{45} + ( -99 - 24 \beta_{1} - 72 \beta_{2} + 2 \beta_{3} - 45 \beta_{4} - 4 \beta_{5} + 26 \beta_{6} + 10 \beta_{7} + 34 \beta_{8} + 9 \beta_{9} + 20 \beta_{10} - 19 \beta_{11} - 31 \beta_{12} + 11 \beta_{13} - 8 \beta_{14} + 2 \beta_{15} ) q^{46} + ( 37 + 6 \beta_{1} - 102 \beta_{2} + 195 \beta_{3} + 28 \beta_{4} - 29 \beta_{5} + 36 \beta_{6} - 8 \beta_{7} - 35 \beta_{8} - 16 \beta_{9} - 25 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} - 10 \beta_{13} - 4 \beta_{14} + 11 \beta_{15} ) q^{47} + ( -220 + 23 \beta_{1} - 15 \beta_{2} + 123 \beta_{3} + 29 \beta_{4} - 6 \beta_{5} + 30 \beta_{6} - 8 \beta_{7} + \beta_{8} + \beta_{9} - 3 \beta_{10} - 9 \beta_{11} + 12 \beta_{12} + 19 \beta_{13} + 6 \beta_{14} + 18 \beta_{15} ) q^{48} + ( -275 - 56 \beta_{1} - 4 \beta_{2} - 168 \beta_{3} + 24 \beta_{4} + 72 \beta_{5} + 68 \beta_{6} - 8 \beta_{7} + 32 \beta_{8} + 8 \beta_{9} + 20 \beta_{10} - 52 \beta_{11} + 4 \beta_{12} - 8 \beta_{13} + 44 \beta_{14} + 24 \beta_{15} ) q^{49} + ( -125 - 125 \beta_{3} ) q^{50} + ( -18 + 4 \beta_{1} + 14 \beta_{2} - 135 \beta_{3} - 5 \beta_{4} + 27 \beta_{5} - 40 \beta_{7} - 31 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} + 27 \beta_{12} + 5 \beta_{13} + 18 \beta_{14} + 9 \beta_{15} ) q^{51} + ( 758 - 4 \beta_{1} + 16 \beta_{2} - 32 \beta_{3} + 12 \beta_{4} + 2 \beta_{5} + 56 \beta_{6} - 40 \beta_{7} - 70 \beta_{8} - 2 \beta_{9} + 18 \beta_{10} - 2 \beta_{11} - 12 \beta_{12} - 18 \beta_{13} + 48 \beta_{14} - 16 \beta_{15} ) q^{52} + ( -360 - 20 \beta_{1} - 2 \beta_{2} + 68 \beta_{3} - 52 \beta_{4} - 30 \beta_{5} - 6 \beta_{6} - 58 \beta_{7} - 78 \beta_{8} - 68 \beta_{9} + 32 \beta_{10} - 24 \beta_{11} - 4 \beta_{12} + 2 \beta_{13} - 14 \beta_{14} - 66 \beta_{15} ) q^{53} + ( -27 + 27 \beta_{2} + 27 \beta_{9} ) q^{54} + ( -9 - 16 \beta_{1} + 7 \beta_{2} - 75 \beta_{3} - 9 \beta_{4} - 6 \beta_{5} - 38 \beta_{6} + 38 \beta_{7} + 39 \beta_{8} + 38 \beta_{9} + 8 \beta_{10} - 15 \beta_{11} + 7 \beta_{12} - 12 \beta_{13} - 14 \beta_{14} - 30 \beta_{15} ) q^{55} + ( -174 + 22 \beta_{1} + 116 \beta_{2} + 70 \beta_{3} + 8 \beta_{4} - 16 \beta_{5} + 24 \beta_{6} - 94 \beta_{7} - 2 \beta_{8} - 20 \beta_{9} + 36 \beta_{10} - 2 \beta_{11} - 40 \beta_{12} - 36 \beta_{13} + 42 \beta_{14} + 72 \beta_{15} ) q^{56} + ( -727 + 22 \beta_{1} - 7 \beta_{2} - 60 \beta_{3} - 23 \beta_{4} - 27 \beta_{5} + 3 \beta_{6} + 35 \beta_{7} + 98 \beta_{8} - 11 \beta_{9} - 12 \beta_{10} - 27 \beta_{11} - 3 \beta_{12} - 4 \beta_{13} - 3 \beta_{14} + 9 \beta_{15} ) q^{57} + ( 838 + 56 \beta_{1} + 40 \beta_{2} + 180 \beta_{3} + 28 \beta_{4} - 58 \beta_{5} + 80 \beta_{6} + 82 \beta_{7} + 50 \beta_{8} - 44 \beta_{9} - 26 \beta_{10} + 36 \beta_{11} - 4 \beta_{12} + 108 \beta_{13} + 24 \beta_{14} + 40 \beta_{15} ) q^{58} + ( 15 - 44 \beta_{1} + 99 \beta_{2} - 51 \beta_{3} + 9 \beta_{4} - 64 \beta_{5} - 42 \beta_{6} + 156 \beta_{7} + 67 \beta_{8} + 36 \beta_{9} - 30 \beta_{10} - 23 \beta_{11} + 55 \beta_{12} - 20 \beta_{13} - 34 \beta_{14} - 72 \beta_{15} ) q^{59} + ( 203 + \beta_{1} + 44 \beta_{2} - 15 \beta_{3} + 21 \beta_{4} - \beta_{5} - 10 \beta_{6} + 5 \beta_{7} - 21 \beta_{10} + 15 \beta_{11} + 5 \beta_{12} + 14 \beta_{13} - 5 \beta_{14} - 10 \beta_{15} ) q^{60} + ( -166 + 76 \beta_{1} + 14 \beta_{2} + 228 \beta_{3} - 44 \beta_{4} - 42 \beta_{5} - 14 \beta_{6} + 154 \beta_{7} + 20 \beta_{8} + 44 \beta_{9} - 36 \beta_{10} - 28 \beta_{11} + 32 \beta_{12} - 18 \beta_{13} - 30 \beta_{14} - 22 \beta_{15} ) q^{61} + ( -973 - 4 \beta_{1} + 132 \beta_{2} + 94 \beta_{3} - 47 \beta_{4} + 12 \beta_{5} - 46 \beta_{6} + 22 \beta_{7} - 34 \beta_{8} + 59 \beta_{9} + 20 \beta_{10} - 13 \beta_{11} - 13 \beta_{12} + 17 \beta_{13} + 28 \beta_{14} + 18 \beta_{15} ) q^{62} + ( 27 + 81 \beta_{3} + 27 \beta_{5} - 54 \beta_{7} - 27 \beta_{8} + 27 \beta_{10} - 27 \beta_{11} + 27 \beta_{15} ) q^{63} + ( -843 - 38 \beta_{1} + 65 \beta_{2} - 520 \beta_{3} + 83 \beta_{4} + 16 \beta_{5} + 42 \beta_{6} - 55 \beta_{7} - 226 \beta_{8} + 55 \beta_{9} + 7 \beta_{10} + 40 \beta_{12} - 23 \beta_{13} + 49 \beta_{14} - 2 \beta_{15} ) q^{64} + ( 131 - 62 \beta_{1} - 13 \beta_{2} - 60 \beta_{3} + 8 \beta_{4} + 48 \beta_{5} + 25 \beta_{6} - 46 \beta_{7} - 36 \beta_{8} - 20 \beta_{9} + 23 \beta_{10} - 25 \beta_{11} - 5 \beta_{12} - 8 \beta_{13} + 35 \beta_{14} ) q^{65} + ( -202 - 60 \beta_{1} + 60 \beta_{2} - 78 \beta_{3} + 26 \beta_{4} + 14 \beta_{5} + 80 \beta_{6} - 66 \beta_{7} - 194 \beta_{8} + 6 \beta_{9} + 6 \beta_{10} - 18 \beta_{11} + 26 \beta_{12} + 42 \beta_{13} + 76 \beta_{14} - 4 \beta_{15} ) q^{66} + ( 34 + 12 \beta_{1} + 264 \beta_{2} - 32 \beta_{3} + 2 \beta_{4} + 68 \beta_{5} + 32 \beta_{6} - 48 \beta_{7} - 68 \beta_{8} - 102 \beta_{9} + 40 \beta_{10} - 30 \beta_{11} + 54 \beta_{12} + 50 \beta_{13} + 28 \beta_{14} + 72 \beta_{15} ) q^{67} + ( -846 + 136 \beta_{1} - 288 \beta_{2} - 204 \beta_{3} + 20 \beta_{4} - 62 \beta_{5} + 56 \beta_{6} + 92 \beta_{7} + 86 \beta_{8} - 98 \beta_{9} - 30 \beta_{10} + 42 \beta_{11} - 28 \beta_{12} + 46 \beta_{13} + 12 \beta_{14} + 32 \beta_{15} ) q^{68} + ( 596 + 10 \beta_{1} - 13 \beta_{2} - 132 \beta_{3} + 13 \beta_{4} - 33 \beta_{6} + 26 \beta_{7} - 16 \beta_{8} + 7 \beta_{9} - 45 \beta_{10} + 6 \beta_{11} - 12 \beta_{12} - \beta_{13} - 9 \beta_{14} + 30 \beta_{15} ) q^{69} + ( -1041 - 14 \beta_{1} - 142 \beta_{2} + 30 \beta_{3} + 39 \beta_{4} + 26 \beta_{5} + 38 \beta_{6} + 42 \beta_{7} + 14 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 25 \beta_{11} + 3 \beta_{12} + 17 \beta_{13} + 14 \beta_{14} + 20 \beta_{15} ) q^{70} + ( 64 + 60 \beta_{1} + 36 \beta_{2} + 244 \beta_{3} + 72 \beta_{4} - 44 \beta_{5} + 112 \beta_{6} + 48 \beta_{7} - 96 \beta_{8} - 192 \beta_{9} - 52 \beta_{10} + 40 \beta_{11} + 56 \beta_{12} - 8 \beta_{13} + 8 \beta_{14} + 60 \beta_{15} ) q^{71} + ( 297 - 27 \beta_{1} - 135 \beta_{2} + 27 \beta_{3} + 27 \beta_{8} + 27 \beta_{10} - 27 \beta_{12} ) q^{72} + ( 658 + 52 \beta_{1} - 22 \beta_{2} + 124 \beta_{3} - 108 \beta_{4} - 118 \beta_{5} - 130 \beta_{6} + 30 \beta_{7} + 220 \beta_{8} - 92 \beta_{9} - 60 \beta_{10} - 12 \beta_{11} - 16 \beta_{12} - 6 \beta_{13} - 82 \beta_{14} - 90 \beta_{15} ) q^{73} + ( 1745 - 52 \beta_{1} + 20 \beta_{2} - 666 \beta_{3} - 23 \beta_{4} - 46 \beta_{5} - 54 \beta_{6} + 208 \beta_{7} + 184 \beta_{8} + 23 \beta_{9} - 42 \beta_{10} + 43 \beta_{11} - 13 \beta_{12} - 15 \beta_{13} + 8 \beta_{14} - 50 \beta_{15} ) q^{74} + 125 \beta_{2} q^{75} + ( -288 + 16 \beta_{1} + 44 \beta_{2} + 160 \beta_{3} + 12 \beta_{4} - 52 \beta_{5} - 120 \beta_{6} + 12 \beta_{7} - 156 \beta_{8} + 104 \beta_{9} - 128 \beta_{10} + 4 \beta_{11} - 40 \beta_{12} + 80 \beta_{13} - 68 \beta_{14} - 120 \beta_{15} ) q^{76} + ( -1474 + 16 \beta_{1} + 60 \beta_{2} + 580 \beta_{3} + 36 \beta_{4} + 102 \beta_{5} + 4 \beta_{6} + 174 \beta_{7} - 136 \beta_{8} + 156 \beta_{9} - 22 \beta_{10} + 18 \beta_{11} + 86 \beta_{12} - 26 \beta_{13} - 24 \beta_{14} - 70 \beta_{15} ) q^{77} + ( -219 + 26 \beta_{1} - 36 \beta_{2} + 102 \beta_{3} - 17 \beta_{4} - 46 \beta_{5} - 10 \beta_{6} + 106 \beta_{7} - 34 \beta_{8} - \beta_{9} - 30 \beta_{10} + 87 \beta_{11} - 13 \beta_{12} - 47 \beta_{13} - 50 \beta_{14} - 28 \beta_{15} ) q^{78} + ( -98 + 32 \beta_{1} + 139 \beta_{2} - 594 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} - 102 \beta_{6} - 24 \beta_{7} - 28 \beta_{8} + 42 \beta_{9} + 19 \beta_{10} + 18 \beta_{11} + 139 \beta_{12} - 192 \beta_{13} - 22 \beta_{14} - 83 \beta_{15} ) q^{79} + ( 688 - 3 \beta_{1} + 131 \beta_{2} - 95 \beta_{3} + 43 \beta_{4} + 62 \beta_{5} + 26 \beta_{6} - 56 \beta_{7} - 85 \beta_{8} - \beta_{9} + 39 \beta_{10} - 15 \beta_{11} + 16 \beta_{12} - 11 \beta_{13} + 58 \beta_{14} + 70 \beta_{15} ) q^{80} + 729 q^{81} + ( -3440 - 88 \beta_{1} + 520 \beta_{2} - 70 \beta_{3} - 2 \beta_{4} + 40 \beta_{5} - 4 \beta_{6} - 188 \beta_{7} + 276 \beta_{8} - 14 \beta_{9} + 90 \beta_{11} - 70 \beta_{12} + 30 \beta_{13} + 48 \beta_{14} + 4 \beta_{15} ) q^{82} + ( 4 - 168 \beta_{1} + 214 \beta_{2} + 190 \beta_{3} - 104 \beta_{4} - 96 \beta_{5} + 4 \beta_{6} + 52 \beta_{7} + 82 \beta_{8} + 170 \beta_{9} - 80 \beta_{10} + 16 \beta_{11} - 12 \beta_{12} + 38 \beta_{13} - 16 \beta_{14} - 76 \beta_{15} ) q^{83} + ( -154 - 8 \beta_{1} - 40 \beta_{2} - 18 \beta_{3} - 4 \beta_{4} - 38 \beta_{5} + 40 \beta_{6} + 110 \beta_{7} + 142 \beta_{8} + 62 \beta_{9} - 30 \beta_{10} + 18 \beta_{11} + 4 \beta_{12} - 34 \beta_{13} + 20 \beta_{14} - 80 \beta_{15} ) q^{84} + ( -690 - 30 \beta_{1} - 15 \beta_{2} - 10 \beta_{3} + 25 \beta_{5} - 105 \beta_{6} - 5 \beta_{7} + 40 \beta_{8} - 20 \beta_{9} - 70 \beta_{10} + 20 \beta_{11} - 10 \beta_{12} - 5 \beta_{13} - 45 \beta_{14} - 45 \beta_{15} ) q^{85} + ( 2380 - 40 \beta_{1} - 180 \beta_{2} - 160 \beta_{3} - 56 \beta_{4} + 48 \beta_{5} - 24 \beta_{6} - 320 \beta_{7} - 64 \beta_{8} - 52 \beta_{9} + 48 \beta_{10} - 64 \beta_{11} - 104 \beta_{12} + 88 \beta_{13} + 56 \beta_{14} + 72 \beta_{15} ) q^{86} + ( -12 + 46 \beta_{1} - 284 \beta_{2} + 138 \beta_{3} - 50 \beta_{4} - 60 \beta_{6} + 170 \beta_{7} + 200 \beta_{8} - 78 \beta_{9} + 72 \beta_{10} + 24 \beta_{11} - 120 \beta_{12} - 28 \beta_{13} - 72 \beta_{14} + 12 \beta_{15} ) q^{87} + ( -290 + 126 \beta_{1} - 336 \beta_{2} + 646 \beta_{3} - 4 \beta_{4} - 32 \beta_{5} + 80 \beta_{6} - 34 \beta_{7} - 442 \beta_{8} - 96 \beta_{9} + 120 \beta_{10} + 94 \beta_{11} + 24 \beta_{12} - 88 \beta_{13} + 22 \beta_{14} + 176 \beta_{15} ) q^{88} + ( 306 - 20 \beta_{1} + 90 \beta_{2} - 608 \beta_{3} + 52 \beta_{4} + 78 \beta_{5} + 114 \beta_{6} - 110 \beta_{7} + 32 \beta_{8} - 16 \beta_{9} + 20 \beta_{10} - 52 \beta_{11} + 16 \beta_{12} + 74 \beta_{13} - 102 \beta_{14} - 86 \beta_{15} ) q^{89} + ( -81 + 27 \beta_{5} - 27 \beta_{7} - 27 \beta_{8} + 27 \beta_{10} ) q^{90} + ( 206 + 40 \beta_{1} - 1216 \beta_{2} + 778 \beta_{3} + 36 \beta_{4} + 26 \beta_{5} + 72 \beta_{6} + 152 \beta_{7} + 42 \beta_{8} - 268 \beta_{9} + 82 \beta_{10} - 78 \beta_{11} - 40 \beta_{12} + 80 \beta_{13} - 56 \beta_{14} + 154 \beta_{15} ) q^{91} + ( 1828 + 8 \beta_{1} - 356 \beta_{2} + 52 \beta_{3} - 32 \beta_{4} + 132 \beta_{5} - 128 \beta_{6} - 128 \beta_{7} + 92 \beta_{8} + 188 \beta_{9} - 24 \beta_{10} - 32 \beta_{11} - 108 \beta_{12} + 12 \beta_{13} - 12 \beta_{14} ) q^{92} + ( 1234 - 18 \beta_{1} - 25 \beta_{2} + 138 \beta_{3} - 82 \beta_{4} - 25 \beta_{5} + 29 \beta_{6} + 45 \beta_{7} + 16 \beta_{8} - 50 \beta_{9} + 30 \beta_{10} - 90 \beta_{11} + 8 \beta_{12} - 33 \beta_{13} + 37 \beta_{14} - 31 \beta_{15} ) q^{93} + ( 3063 - 52 \beta_{1} + 100 \beta_{2} - 410 \beta_{3} + 93 \beta_{4} + 56 \beta_{5} + 102 \beta_{6} - 330 \beta_{7} + 70 \beta_{8} + 95 \beta_{9} + 64 \beta_{10} - 129 \beta_{11} + 99 \beta_{12} + 17 \beta_{13} + 68 \beta_{14} + 82 \beta_{15} ) q^{94} + ( 17 - 22 \beta_{1} + 354 \beta_{2} + 85 \beta_{3} + 22 \beta_{4} + 33 \beta_{5} + 4 \beta_{6} - 224 \beta_{7} - 117 \beta_{8} + 146 \beta_{9} + \beta_{10} - 45 \beta_{11} - 16 \beta_{12} - 4 \beta_{13} + 32 \beta_{14} + 5 \beta_{15} ) q^{95} + ( 1185 - 60 \beta_{1} + 487 \beta_{2} + 114 \beta_{3} - 27 \beta_{4} + 36 \beta_{5} + 6 \beta_{6} - 111 \beta_{7} + 84 \beta_{8} - 15 \beta_{9} + 45 \beta_{10} + 30 \beta_{11} + 12 \beta_{12} - 57 \beta_{13} - 27 \beta_{14} - 30 \beta_{15} ) q^{96} + ( -832 - 88 \beta_{1} - 56 \beta_{2} + 276 \beta_{3} + 52 \beta_{4} + 18 \beta_{5} - 72 \beta_{6} - 166 \beta_{7} + 160 \beta_{8} - 20 \beta_{9} + 50 \beta_{10} + 130 \beta_{11} - 42 \beta_{12} - 14 \beta_{13} + 92 \beta_{14} + 62 \beta_{15} ) q^{97} + ( 1619 + 352 \beta_{1} - 64 \beta_{2} + 383 \beta_{3} + 44 \beta_{4} + 56 \beta_{5} + 104 \beta_{6} + 32 \beta_{7} - 64 \beta_{8} - 140 \beta_{9} + 104 \beta_{10} - 60 \beta_{11} - 44 \beta_{12} + 44 \beta_{13} - 144 \beta_{14} + 232 \beta_{15} ) q^{98} + ( -27 - 27 \beta_{2} + 27 \beta_{3} - 81 \beta_{4} - 54 \beta_{6} + 81 \beta_{8} + 54 \beta_{10} + 27 \beta_{11} - 27 \beta_{12} - 108 \beta_{13} - 54 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 12q^{2} + 26q^{4} + 18q^{6} - 180q^{8} - 432q^{9} + O(q^{10})$$ $$16q - 12q^{2} + 26q^{4} + 18q^{6} - 180q^{8} - 432q^{9} + 50q^{10} - 352q^{13} - 804q^{14} - 190q^{16} + 324q^{18} + 600q^{20} + 288q^{21} + 436q^{22} - 1998q^{24} + 2000q^{25} - 852q^{26} - 1156q^{28} - 3456q^{29} + 7668q^{32} + 4772q^{34} - 702q^{36} + 9376q^{37} - 1320q^{38} + 550q^{40} + 1248q^{41} - 324q^{42} - 6420q^{44} - 1112q^{46} - 4176q^{48} - 3952q^{49} - 1500q^{50} + 12704q^{52} - 5184q^{53} - 486q^{54} - 2604q^{56} - 11232q^{57} + 12708q^{58} + 3150q^{60} - 3808q^{61} - 16152q^{62} - 11902q^{64} + 2400q^{65} - 2916q^{66} - 12312q^{68} + 9792q^{69} - 17100q^{70} + 4860q^{72} + 11040q^{73} + 30516q^{74} - 5160q^{76} - 27456q^{77} - 3600q^{78} + 10800q^{80} + 11664q^{81} - 54040q^{82} - 2052q^{84} - 11200q^{85} + 39768q^{86} - 7220q^{88} + 7584q^{89} - 1350q^{90} + 28848q^{92} + 19872q^{93} + 49776q^{94} + 18882q^{96} - 14496q^{97} + 23940q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} - 9 x^{14} + 18 x^{13} + 263 x^{12} - 444 x^{11} - 1732 x^{10} - 832 x^{9} + 25296 x^{8} - 6656 x^{7} - 110848 x^{6} - 227328 x^{5} + 1077248 x^{4} + 589824 x^{3} - 2359296 x^{2} - 8388608 x + 16777216$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-11071 \nu^{15} + 226436 \nu^{14} + 593367 \nu^{13} - 554294 \nu^{12} - 21258985 \nu^{11} + 10003324 \nu^{10} + 130862492 \nu^{9} + 322351008 \nu^{8} - 1602733616 \nu^{7} - 2121919360 \nu^{6} + 3116464896 \nu^{5} + 34170421248 \nu^{4} - 26216960000 \nu^{3} - 124762554368 \nu^{2} - 91851063296 \nu + 741363154944$$$$)/ 15454961664$$ $$\beta_{2}$$ $$=$$ $$($$$$-6617 \nu^{15} + 26278 \nu^{14} + 88297 \nu^{13} + 60460 \nu^{12} - 2080027 \nu^{11} + 578762 \nu^{10} + 12589276 \nu^{9} + 31545336 \nu^{8} - 152490000 \nu^{7} - 165011552 \nu^{6} + 471993856 \nu^{5} + 2816759296 \nu^{4} - 3647557632 \nu^{3} - 10337263616 \nu^{2} - 3917545472 \nu + 64036536320$$$$)/ 2575826944$$ $$\beta_{3}$$ $$=$$ $$($$$$37951 \nu^{15} - 50364 \nu^{14} - 476215 \nu^{13} - 676818 \nu^{12} + 8411705 \nu^{11} + 6300796 \nu^{10} - 49013820 \nu^{9} - 178779072 \nu^{8} + 490096944 \nu^{7} + 1176020480 \nu^{6} - 1041215232 \nu^{5} - 12046526464 \nu^{4} + 6948376576 \nu^{3} + 44963463168 \nu^{2} + 46226472960 \nu - 214580068352$$$$)/ 7727480832$$ $$\beta_{4}$$ $$=$$ $$($$$$-40727 \nu^{15} - 80592 \nu^{14} - 189281 \nu^{13} + 3024494 \nu^{12} + 5984551 \nu^{11} - 26863120 \nu^{10} - 117430068 \nu^{9} + 117938928 \nu^{8} + 1121548752 \nu^{7} - 193992128 \nu^{6} - 8485902080 \nu^{5} - 7644802048 \nu^{4} + 49677971456 \nu^{3} + 88222842880 \nu^{2} - 152971902976 \nu - 583508426752$$$$)/ 7727480832$$ $$\beta_{5}$$ $$=$$ $$($$$$99391 \nu^{15} - 371968 \nu^{14} - 502471 \nu^{13} + 212754 \nu^{12} + 14496273 \nu^{11} - 20667552 \nu^{10} - 42012588 \nu^{9} - 179603504 \nu^{8} + 1153217712 \nu^{7} - 328005952 \nu^{6} - 778729728 \nu^{5} - 18357978112 \nu^{4} + 23344009216 \nu^{3} + 4840308736 \nu^{2} + 222821744640 \nu - 357198462976$$$$)/ 15454961664$$ $$\beta_{6}$$ $$=$$ $$($$$$-18263 \nu^{15} + 54973 \nu^{14} + 350339 \nu^{13} - 167015 \nu^{12} - 5995975 \nu^{11} + 538875 \nu^{10} + 52716712 \nu^{9} + 73912668 \nu^{8} - 468398416 \nu^{7} - 729702192 \nu^{6} + 2317975680 \nu^{5} + 8375254784 \nu^{4} - 12463949824 \nu^{3} - 43117391872 \nu^{2} + 12147032064 \nu + 215238705152$$$$)/ 965935104$$ $$\beta_{7}$$ $$=$$ $$($$$$158297 \nu^{15} - 321188 \nu^{14} - 2321441 \nu^{13} - 1304702 \nu^{12} + 40098543 \nu^{11} + 18285284 \nu^{10} - 287175460 \nu^{9} - 710621760 \nu^{8} + 2661758544 \nu^{7} + 5196978688 \nu^{6} - 8932489472 \nu^{5} - 56814786560 \nu^{4} + 50861223936 \nu^{3} + 239692152832 \nu^{2} + 101691949056 \nu - 1177881673728$$$$)/ 7727480832$$ $$\beta_{8}$$ $$=$$ $$($$$$-9755 \nu^{15} + 13120 \nu^{14} + 127395 \nu^{13} + 144550 \nu^{12} - 2087125 \nu^{11} - 1558880 \nu^{10} + 13170940 \nu^{9} + 40784240 \nu^{8} - 118379760 \nu^{7} - 283405760 \nu^{6} + 317326080 \nu^{5} + 2821985280 \nu^{4} - 1776168960 \nu^{3} - 11276861440 \nu^{2} - 7274496000 \nu + 48431104000$$$$)/ 454557696$$ $$\beta_{9}$$ $$=$$ $$($$$$28268 \nu^{15} - 40905 \nu^{14} - 260496 \nu^{13} - 675047 \nu^{12} + 5667738 \nu^{11} + 3456561 \nu^{10} - 24303468 \nu^{9} - 137186940 \nu^{8} + 318838944 \nu^{7} + 662545328 \nu^{6} - 86605440 \nu^{5} - 8489875200 \nu^{4} + 4521709568 \nu^{3} + 24414425088 \nu^{2} + 39889698816 \nu - 150021603328$$$$)/ 1287913472$$ $$\beta_{10}$$ $$=$$ $$($$$$388523 \nu^{15} - 954528 \nu^{14} - 7727251 \nu^{13} - 4008678 \nu^{12} + 134476773 \nu^{11} + 56582400 \nu^{10} - 965502812 \nu^{9} - 2581133936 \nu^{8} + 9178352496 \nu^{7} + 17916705728 \nu^{6} - 29559931136 \nu^{5} - 210565854208 \nu^{4} + 200635273216 \nu^{3} + 866626715648 \nu^{2} + 475815084032 \nu - 4911664201728$$$$)/ 15454961664$$ $$\beta_{11}$$ $$=$$ $$($$$$155836 \nu^{15} + 5199 \nu^{14} - 2013264 \nu^{13} - 5482783 \nu^{12} + 30058250 \nu^{11} + 55367353 \nu^{10} - 130974524 \nu^{9} - 911580188 \nu^{8} + 1258153952 \nu^{7} + 5439211312 \nu^{6} + 2208130176 \nu^{5} - 51319926528 \nu^{4} - 4952768512 \nu^{3} + 150888419328 \nu^{2} + 327351762944 \nu - 756971995136$$$$)/ 3863740416$$ $$\beta_{12}$$ $$=$$ $$($$$$39828 \nu^{15} - 39001 \nu^{14} - 561192 \nu^{13} - 949223 \nu^{12} + 8771394 \nu^{11} + 10298721 \nu^{10} - 50310748 \nu^{9} - 213246844 \nu^{8} + 474350048 \nu^{7} + 1429650608 \nu^{6} - 694801792 \nu^{5} - 13919221504 \nu^{4} + 4471156736 \nu^{3} + 52419858432 \nu^{2} + 56009785344 \nu - 246523756544$$$$)/ 965935104$$ $$\beta_{13}$$ $$=$$ $$($$$$322639 \nu^{15} - 355100 \nu^{14} - 4508775 \nu^{13} - 7378706 \nu^{12} + 75683209 \nu^{11} + 72681372 \nu^{10} - 446839260 \nu^{9} - 1747727296 \nu^{8} + 4364619440 \nu^{7} + 11364439040 \nu^{6} - 7141947648 \nu^{5} - 119140616192 \nu^{4} + 65090646016 \nu^{3} + 442039140352 \nu^{2} + 475082653696 \nu - 2320671178752$$$$)/ 7727480832$$ $$\beta_{14}$$ $$=$$ $$($$$$-742535 \nu^{15} + 988864 \nu^{14} + 9060815 \nu^{13} + 15925822 \nu^{12} - 153874377 \nu^{11} - 124918816 \nu^{10} + 781890124 \nu^{9} + 3528082352 \nu^{8} - 8205853488 \nu^{7} - 20453307584 \nu^{6} + 6044823808 \nu^{5} + 218569614336 \nu^{4} - 80455061504 \nu^{3} - 691300417536 \nu^{2} - 1013217361920 \nu + 3457620639744$$$$)/ 15454961664$$ $$\beta_{15}$$ $$=$$ $$($$$$383855 \nu^{15} - 709602 \nu^{14} - 4960223 \nu^{13} - 4710044 \nu^{12} + 88176749 \nu^{11} + 37865554 \nu^{10} - 549147140 \nu^{9} - 1636062184 \nu^{8} + 5592759280 \nu^{7} + 10935769888 \nu^{6} - 15134208000 \nu^{5} - 122897572352 \nu^{4} + 101804773376 \nu^{3} + 461909090304 \nu^{2} + 242148769792 \nu - 2323994116096$$$$)/ 7727480832$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-5 \beta_{15} - \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 3 \beta_{9} + \beta_{8} - \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - \beta_{4} + 45 \beta_{3} - 7 \beta_{2} + \beta_{1} + 40$$$$)/120$$ $$\nu^{2}$$ $$=$$ $$($$$$-10 \beta_{15} + 4 \beta_{14} + 9 \beta_{13} + 23 \beta_{12} - 15 \beta_{11} - 6 \beta_{10} + 7 \beta_{9} - 38 \beta_{8} - 18 \beta_{7} + 8 \beta_{6} + 8 \beta_{5} + 17 \beta_{4} - 60 \beta_{3} - 6 \beta_{1} + 253$$$$)/120$$ $$\nu^{3}$$ $$=$$ $$($$$$10 \beta_{15} + 5 \beta_{14} + 21 \beta_{13} - 10 \beta_{12} - 10 \beta_{11} + 9 \beta_{10} - 5 \beta_{9} - 8 \beta_{8} - 18 \beta_{7} + 10 \beta_{6} - \beta_{5} - 6 \beta_{4} + 26 \beta_{2} + 9 \beta_{1} + 288$$$$)/40$$ $$\nu^{4}$$ $$=$$ $$($$$$-10 \beta_{15} - 59 \beta_{14} + 72 \beta_{13} + 17 \beta_{12} - 105 \beta_{11} - 15 \beta_{10} - 82 \beta_{9} + 52 \beta_{8} + 30 \beta_{7} + 122 \beta_{6} - 79 \beta_{5} + 77 \beta_{4} + 810 \beta_{3} - 64 \beta_{2} + 75 \beta_{1} - 2415$$$$)/120$$ $$\nu^{5}$$ $$=$$ $$($$$$24 \beta_{15} - 30 \beta_{14} + 103 \beta_{13} - 15 \beta_{12} - 45 \beta_{11} - 30 \beta_{10} - 21 \beta_{9} + 4 \beta_{8} + 28 \beta_{7} + 36 \beta_{6} - 12 \beta_{5} + 17 \beta_{4} - 498 \beta_{3} + 198 \beta_{2} - 40 \beta_{1} + 231$$$$)/24$$ $$\nu^{6}$$ $$=$$ $$($$$$130 \beta_{15} - 135 \beta_{14} + 159 \beta_{13} - 140 \beta_{12} - 120 \beta_{11} + 5 \beta_{10} - 195 \beta_{9} + 744 \beta_{8} - 226 \beta_{7} - 190 \beta_{6} - 185 \beta_{5} - 304 \beta_{4} + 2740 \beta_{3} + 654 \beta_{2} + \beta_{1} - 2150$$$$)/40$$ $$\nu^{7}$$ $$=$$ $$($$$$2130 \beta_{15} - 561 \beta_{14} + 1342 \beta_{13} + 873 \beta_{12} - 225 \beta_{11} - 2445 \beta_{10} - 2888 \beta_{9} + 6676 \beta_{8} + 4474 \beta_{7} + 2718 \beta_{6} - 681 \beta_{5} + 3029 \beta_{4} + 4770 \beta_{3} - 3692 \beta_{2} + 581 \beta_{1} + 8765$$$$)/120$$ $$\nu^{8}$$ $$=$$ $$($$$$2660 \beta_{15} + 2668 \beta_{14} + 9183 \beta_{13} - 3349 \beta_{12} + 1305 \beta_{11} - 5292 \beta_{10} - 8561 \beta_{9} - 4676 \beta_{8} - 5736 \beta_{7} - 1864 \beta_{6} + 3686 \beta_{5} - 2221 \beta_{4} - 17310 \beta_{3} - 17690 \beta_{2} - 7122 \beta_{1} - 59759$$$$)/120$$ $$\nu^{9}$$ $$=$$ $$($$$$10170 \beta_{15} + 7745 \beta_{14} - 531 \beta_{13} + 7090 \beta_{12} - 2870 \beta_{11} + 2821 \beta_{10} - 5765 \beta_{9} - 9032 \beta_{8} - 19562 \beta_{7} + 9410 \beta_{6} + 4331 \beta_{5} + 5326 \beta_{4} + 29280 \beta_{3} - 3526 \beta_{2} - 419 \beta_{1} - 54628$$$$)/40$$ $$\nu^{10}$$ $$=$$ $$($$$$14382 \beta_{15} + 16053 \beta_{14} + 10384 \beta_{13} - 207 \beta_{12} - 2121 \beta_{11} + 1161 \beta_{10} - 11826 \beta_{9} - 15044 \beta_{8} + 17830 \beta_{7} + 28602 \beta_{6} + 1017 \beta_{5} + 5381 \beta_{4} - 48174 \beta_{3} - 23128 \beta_{2} + 779 \beta_{1} - 47583$$$$)/24$$ $$\nu^{11}$$ $$=$$ $$($$$$83800 \beta_{15} + 168842 \beta_{14} + 46919 \beta_{13} - 70151 \beta_{12} - 16005 \beta_{11} + 82602 \beta_{10} - 99669 \beta_{9} - 681404 \beta_{8} - 409492 \beta_{7} + 176404 \beta_{6} + 25564 \beta_{5} + 21929 \beta_{4} + 365790 \beta_{3} - 290698 \beta_{2} - 206480 \beta_{1} - 4468481$$$$)/120$$ $$\nu^{12}$$ $$=$$ $$($$$$208250 \beta_{15} + 175845 \beta_{14} - 49381 \beta_{13} + 70700 \beta_{12} - 83200 \beta_{11} + 159745 \beta_{10} + 41185 \beta_{9} - 323416 \beta_{8} - 77466 \beta_{7} + 296410 \beta_{6} + 117675 \beta_{5} + 99896 \beta_{4} - 243180 \beta_{3} + 1230694 \beta_{2} - 87139 \beta_{1} - 535350$$$$)/40$$ $$\nu^{13}$$ $$=$$ $$($$$$-565350 \beta_{15} + 969483 \beta_{14} + 46014 \beta_{13} - 974619 \beta_{12} + 197715 \beta_{11} - 279345 \beta_{10} + 1567744 \beta_{9} + 108852 \beta_{8} + 2739378 \beta_{7} + 11766 \beta_{6} - 834957 \beta_{5} - 970527 \beta_{4} + 5361450 \beta_{3} + 1097076 \beta_{2} - 1046103 \beta_{1} - 6725815$$$$)/120$$ $$\nu^{14}$$ $$=$$ $$($$$$780740 \beta_{15} + 820372 \beta_{14} - 1525773 \beta_{13} - 1778281 \beta_{12} + 3214605 \beta_{11} - 1238388 \beta_{10} + 1283411 \beta_{9} - 4531364 \beta_{8} - 5516424 \beta_{7} - 1560856 \beta_{6} + 1408214 \beta_{5} + 5108351 \beta_{4} + 9633210 \beta_{3} + 13264910 \beta_{2} - 2579778 \beta_{1} - 5561451$$$$)/120$$ $$\nu^{15}$$ $$=$$ $$($$$$-34870 \beta_{15} + 344889 \beta_{14} - 111235 \beta_{13} + 157722 \beta_{12} + 30306 \beta_{11} - 365027 \beta_{10} + 1162843 \beta_{9} - 372008 \beta_{8} + 338886 \beta_{7} - 502478 \beta_{6} + 719363 \beta_{5} + 72118 \beta_{4} - 1929872 \beta_{3} + 4788154 \beta_{2} + 26917 \beta_{1} - 4002364$$$$)/8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$41$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 2.44021 + 1.43016i 2.44021 − 1.43016i −1.14149 + 2.58786i −1.14149 − 2.58786i 1.85226 − 2.13755i 1.85226 + 2.13755i 2.70166 + 0.837276i 2.70166 − 0.837276i 2.77114 + 0.566380i 2.77114 − 0.566380i −1.85197 + 2.13780i −1.85197 − 2.13780i −2.48191 − 1.35651i −2.48191 + 1.35651i −2.28990 + 1.66022i −2.28990 − 1.66022i
−3.95975 0.566024i 5.19615i 15.3592 + 4.48263i 11.1803 2.94115 20.5755i 24.1355i −58.2814 26.4438i −27.0000 −44.2713 6.32834i
31.2 −3.95975 + 0.566024i 5.19615i 15.3592 4.48263i 11.1803 2.94115 + 20.5755i 24.1355i −58.2814 + 26.4438i −27.0000 −44.2713 + 6.32834i
31.3 −3.40825 2.09376i 5.19615i 7.23235 + 14.2721i −11.1803 −10.8795 + 17.7098i 61.3317i 5.23271 63.7857i −27.0000 38.1054 + 23.4089i
31.4 −3.40825 + 2.09376i 5.19615i 7.23235 14.2721i −11.1803 −10.8795 17.7098i 61.3317i 5.23271 + 63.7857i −27.0000 38.1054 23.4089i
31.5 −3.34902 2.18726i 5.19615i 6.43181 + 14.6503i −11.1803 11.3653 17.4020i 1.39605i 10.5038 63.1322i −27.0000 37.4431 + 24.4543i
31.6 −3.34902 + 2.18726i 5.19615i 6.43181 14.6503i −11.1803 11.3653 + 17.4020i 1.39605i 10.5038 + 63.1322i −27.0000 37.4431 24.4543i
31.7 −3.08838 2.54203i 5.19615i 3.07620 + 15.7015i 11.1803 −13.2088 + 16.0477i 86.5709i 30.4132 56.3120i −27.0000 −34.5292 28.4207i
31.8 −3.08838 + 2.54203i 5.19615i 3.07620 15.7015i 11.1803 −13.2088 16.0477i 86.5709i 30.4132 + 56.3120i −27.0000 −34.5292 + 28.4207i
31.9 0.264404 3.99125i 5.19615i −15.8602 2.11060i −11.1803 20.7392 + 1.37388i 29.5855i −12.6174 + 62.7439i −27.0000 −2.95612 + 44.6236i
31.10 0.264404 + 3.99125i 5.19615i −15.8602 + 2.11060i −11.1803 20.7392 1.37388i 29.5855i −12.6174 62.7439i −27.0000 −2.95612 44.6236i
31.11 1.28004 3.78966i 5.19615i −12.7230 9.70180i 11.1803 19.6916 + 6.65126i 89.0673i −53.0524 + 35.7972i −27.0000 14.3112 42.3697i
31.12 1.28004 + 3.78966i 5.19615i −12.7230 + 9.70180i 11.1803 19.6916 6.65126i 89.0673i −53.0524 35.7972i −27.0000 14.3112 + 42.3697i
31.13 2.37483 3.21872i 5.19615i −4.72038 15.2878i −11.1803 −16.7250 12.3400i 19.2859i −60.4174 21.1124i −27.0000 −26.5514 + 35.9864i
31.14 2.37483 + 3.21872i 5.19615i −4.72038 + 15.2878i −11.1803 −16.7250 + 12.3400i 19.2859i −60.4174 + 21.1124i −27.0000 −26.5514 35.9864i
31.15 3.88613 0.947630i 5.19615i 14.2040 7.36522i 11.1803 −4.92403 20.1929i 12.7755i 48.2191 42.0823i −27.0000 43.4482 10.5948i
31.16 3.88613 + 0.947630i 5.19615i 14.2040 + 7.36522i 11.1803 −4.92403 + 20.1929i 12.7755i 48.2191 + 42.0823i −27.0000 43.4482 + 10.5948i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.5.c.a 16
3.b odd 2 1 180.5.c.c 16
4.b odd 2 1 inner 60.5.c.a 16
5.b even 2 1 300.5.c.d 16
5.c odd 4 2 300.5.f.b 32
8.b even 2 1 960.5.e.f 16
8.d odd 2 1 960.5.e.f 16
12.b even 2 1 180.5.c.c 16
20.d odd 2 1 300.5.c.d 16
20.e even 4 2 300.5.f.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.c.a 16 1.a even 1 1 trivial
60.5.c.a 16 4.b odd 2 1 inner
180.5.c.c 16 3.b odd 2 1
180.5.c.c 16 12.b even 2 1
300.5.c.d 16 5.b even 2 1
300.5.c.d 16 20.d odd 2 1
300.5.f.b 32 5.c odd 4 2
300.5.f.b 32 20.e even 4 2
960.5.e.f 16 8.b even 2 1
960.5.e.f 16 8.d odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4294967296 + 3221225472 T + 989855744 T^{2} + 201326592 T^{3} + 51118080 T^{4} + 7864320 T^{5} - 1347584 T^{6} - 589824 T^{7} - 103936 T^{8} - 36864 T^{9} - 5264 T^{10} + 1920 T^{11} + 780 T^{12} + 192 T^{13} + 59 T^{14} + 12 T^{15} + T^{16}$$
$3$ $$( 27 + T^{2} )^{8}$$
$5$ $$( -125 + T^{2} )^{8}$$
$7$ $$13\!\cdots\!76$$$$+$$$$70\!\cdots\!36$$$$T^{2} + 85125139720803713024 T^{4} + 346323556398792704 T^{6} + 612313337626624 T^{8} + 484373856256 T^{10} + 157121024 T^{12} + 21184 T^{14} + T^{16}$$
$11$ $$15\!\cdots\!56$$$$+$$$$34\!\cdots\!64$$$$T^{2} +$$$$21\!\cdots\!68$$$$T^{4} +$$$$55\!\cdots\!60$$$$T^{6} + 6511204470995746816 T^{8} + 379916641239040 T^{10} + 11469588992 T^{12} + 171328 T^{14} + T^{16}$$
$13$ $$( -236865200916224 + 197808834663424 T + 18366530120960 T^{2} + 457484808448 T^{3} + 1883800288 T^{4} - 19651648 T^{5} - 108400 T^{6} + 176 T^{7} + T^{8} )^{2}$$
$17$ $$( 31387559961760000 + 3402695341056000 T - 101347242579200 T^{2} - 1101359124480 T^{3} + 26518288224 T^{4} + 10007040 T^{5} - 376016 T^{6} + T^{8} )^{2}$$
$19$ $$70\!\cdots\!76$$$$+$$$$11\!\cdots\!56$$$$T^{2} +$$$$80\!\cdots\!24$$$$T^{4} +$$$$20\!\cdots\!84$$$$T^{6} +$$$$26\!\cdots\!64$$$$T^{8} + 178536142820458496 T^{10} + 675039593984 T^{12} + 1308224 T^{14} + T^{16}$$
$23$ $$55\!\cdots\!56$$$$+$$$$10\!\cdots\!08$$$$T^{2} +$$$$34\!\cdots\!68$$$$T^{4} +$$$$24\!\cdots\!40$$$$T^{6} +$$$$46\!\cdots\!56$$$$T^{8} + 361427060279459840 T^{10} + 1275082515968 T^{12} + 1931072 T^{14} + T^{16}$$
$29$ $$($$$$81\!\cdots\!36$$$$-$$$$45\!\cdots\!92$$$$T - 1118291516329069824 T^{2} + 2848989078524928 T^{3} + 2663603833184 T^{4} - 4231564032 T^{5} - 2669904 T^{6} + 1728 T^{7} + T^{8} )^{2}$$
$31$ $$48\!\cdots\!16$$$$+$$$$10\!\cdots\!16$$$$T^{2} +$$$$33\!\cdots\!56$$$$T^{4} +$$$$25\!\cdots\!16$$$$T^{6} +$$$$83\!\cdots\!64$$$$T^{8} + 14181599268396580864 T^{10} + 12694643475968 T^{12} + 5688640 T^{14} + T^{16}$$
$37$ $$( -$$$$27\!\cdots\!04$$$$-$$$$21\!\cdots\!88$$$$T + 6449925012927361280 T^{2} + 2147497175572736 T^{3} - 15073362040352 T^{4} + 8307353536 T^{5} + 4335440 T^{6} - 4688 T^{7} + T^{8} )^{2}$$
$41$ $$($$$$12\!\cdots\!64$$$$+$$$$12\!\cdots\!44$$$$T - 37139646368459191552 T^{2} - 22118152285890816 T^{3} + 39511986540384 T^{4} + 8589432000 T^{5} - 13730512 T^{6} - 624 T^{7} + T^{8} )^{2}$$
$43$ $$10\!\cdots\!76$$$$+$$$$73\!\cdots\!24$$$$T^{2} +$$$$19\!\cdots\!84$$$$T^{4} +$$$$26\!\cdots\!76$$$$T^{6} +$$$$20\!\cdots\!64$$$$T^{8} +$$$$86\!\cdots\!64$$$$T^{10} + 202841606319104 T^{12} + 23359616 T^{14} + T^{16}$$
$47$ $$16\!\cdots\!00$$$$+$$$$27\!\cdots\!00$$$$T^{2} +$$$$13\!\cdots\!00$$$$T^{4} +$$$$27\!\cdots\!00$$$$T^{6} +$$$$27\!\cdots\!96$$$$T^{8} +$$$$13\!\cdots\!88$$$$T^{10} + 323664680027648 T^{12} + 31283648 T^{14} + T^{16}$$
$53$ $$($$$$86\!\cdots\!36$$$$-$$$$38\!\cdots\!88$$$$T -$$$$42\!\cdots\!40$$$$T^{2} + 1235641682182903296 T^{3} + 709905831679328 T^{4} - 108032893824 T^{5} - 46935120 T^{6} + 2592 T^{7} + T^{8} )^{2}$$
$59$ $$44\!\cdots\!36$$$$+$$$$49\!\cdots\!56$$$$T^{2} +$$$$14\!\cdots\!04$$$$T^{4} +$$$$74\!\cdots\!64$$$$T^{6} +$$$$14\!\cdots\!64$$$$T^{8} +$$$$13\!\cdots\!56$$$$T^{10} + 6195985168034304 T^{12} + 128706624 T^{14} + T^{16}$$
$61$ $$( -$$$$20\!\cdots\!24$$$$-$$$$61\!\cdots\!12$$$$T -$$$$41\!\cdots\!48$$$$T^{2} + 924285269314105600 T^{3} + 937740836323936 T^{4} - 73948573120 T^{5} - 55178512 T^{6} + 1904 T^{7} + T^{8} )^{2}$$
$67$ $$57\!\cdots\!76$$$$+$$$$33\!\cdots\!56$$$$T^{2} +$$$$82\!\cdots\!24$$$$T^{4} +$$$$11\!\cdots\!44$$$$T^{6} +$$$$86\!\cdots\!04$$$$T^{8} +$$$$40\!\cdots\!16$$$$T^{10} + 11247396539055104 T^{12} + 165284224 T^{14} + T^{16}$$
$71$ $$21\!\cdots\!00$$$$+$$$$58\!\cdots\!00$$$$T^{2} +$$$$49\!\cdots\!00$$$$T^{4} +$$$$13\!\cdots\!00$$$$T^{6} +$$$$14\!\cdots\!56$$$$T^{8} +$$$$77\!\cdots\!12$$$$T^{10} + 19482745425010688 T^{12} + 227828992 T^{14} + T^{16}$$
$73$ $$( -$$$$68\!\cdots\!96$$$$-$$$$30\!\cdots\!60$$$$T +$$$$48\!\cdots\!24$$$$T^{2} - 20581224701118447360 T^{3} + 1557357338282592 T^{4} + 715952815680 T^{5} - 103764752 T^{6} - 5520 T^{7} + T^{8} )^{2}$$
$79$ $$46\!\cdots\!56$$$$+$$$$28\!\cdots\!80$$$$T^{2} +$$$$98\!\cdots\!88$$$$T^{4} +$$$$13\!\cdots\!64$$$$T^{6} +$$$$82\!\cdots\!84$$$$T^{8} +$$$$25\!\cdots\!16$$$$T^{10} + 41506580262811136 T^{12} + 327654976 T^{14} + T^{16}$$
$83$ $$61\!\cdots\!96$$$$+$$$$11\!\cdots\!20$$$$T^{2} +$$$$92\!\cdots\!68$$$$T^{4} +$$$$39\!\cdots\!96$$$$T^{6} +$$$$96\!\cdots\!64$$$$T^{8} +$$$$14\!\cdots\!04$$$$T^{10} + 120617011204729856 T^{12} + 544982144 T^{14} + T^{16}$$
$89$ $$( -$$$$46\!\cdots\!76$$$$-$$$$21\!\cdots\!12$$$$T -$$$$10\!\cdots\!28$$$$T^{2} + 1094553074551375104 T^{3} + 33177059847968864 T^{4} + 642136991040 T^{5} - 324562320 T^{6} - 3792 T^{7} + T^{8} )^{2}$$
$97$ $$( -$$$$12\!\cdots\!16$$$$+$$$$14\!\cdots\!36$$$$T -$$$$46\!\cdots\!28$$$$T^{2} - 5459530641178662144 T^{3} + 18322782073722720 T^{4} - 784704421440 T^{5} - 233395152 T^{6} + 7248 T^{7} + T^{8} )^{2}$$