# Properties

 Label 17.3.e.b Level 17 Weight 3 Character orbit 17.e Analytic conductor 0.463 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$17$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 17.e (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.463216449413$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{16}^{2} + \zeta_{16}^{4} + \zeta_{16}^{7} ) q^{2} + ( 2 \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{3} + ( -1 + 2 \zeta_{16} - 2 \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{4} + ( -3 - \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{4} - 3 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{5} + ( -5 \zeta_{16} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{4} + 5 \zeta_{16}^{6} ) q^{6} + ( -2 + \zeta_{16} - \zeta_{16}^{2} + 2 \zeta_{16}^{3} - \zeta_{16}^{4} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{7} + ( 2 - 2 \zeta_{16}^{2} - 3 \zeta_{16}^{3} - \zeta_{16}^{4} - \zeta_{16}^{5} - \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{8} + ( 1 + 3 \zeta_{16} + \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 7 \zeta_{16}^{4} - 7 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{16}^{2} + \zeta_{16}^{4} + \zeta_{16}^{7} ) q^{2} + ( 2 \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{3} + ( -1 + 2 \zeta_{16} - 2 \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{4} + ( -3 - \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{4} - 3 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{5} + ( -5 \zeta_{16} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{4} + 5 \zeta_{16}^{6} ) q^{6} + ( -2 + \zeta_{16} - \zeta_{16}^{2} + 2 \zeta_{16}^{3} - \zeta_{16}^{4} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{7} + ( 2 - 2 \zeta_{16}^{2} - 3 \zeta_{16}^{3} - \zeta_{16}^{4} - \zeta_{16}^{5} - \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{8} + ( 1 + 3 \zeta_{16} + \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 7 \zeta_{16}^{4} - 7 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{9} + ( 6 \zeta_{16} + 6 \zeta_{16}^{2} + \zeta_{16}^{4} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{10} + ( 5 + 3 \zeta_{16}^{2} + 3 \zeta_{16}^{5} + 5 \zeta_{16}^{7} ) q^{11} + ( 5 - 4 \zeta_{16} - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{3} + 4 \zeta_{16}^{4} - 5 \zeta_{16}^{5} + 3 \zeta_{16}^{6} + 3 \zeta_{16}^{7} ) q^{12} + ( -2 \zeta_{16}^{2} + 7 \zeta_{16}^{5} - 7 \zeta_{16}^{7} ) q^{13} + ( 2 - 2 \zeta_{16} - 2 \zeta_{16}^{2} + \zeta_{16}^{3} - 3 \zeta_{16}^{4} - 3 \zeta_{16}^{5} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{14} + ( 4 - 7 \zeta_{16} - 10 \zeta_{16}^{2} - 6 \zeta_{16}^{3} - 10 \zeta_{16}^{4} - 7 \zeta_{16}^{5} + 4 \zeta_{16}^{6} ) q^{15} + ( 8 \zeta_{16} - 2 \zeta_{16}^{2} + 4 \zeta_{16}^{3} + \zeta_{16}^{4} + 4 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + 8 \zeta_{16}^{7} ) q^{16} + ( -2 - 9 \zeta_{16} + 4 \zeta_{16}^{2} + 7 \zeta_{16}^{3} + 8 \zeta_{16}^{4} - 5 \zeta_{16}^{5} + 7 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{17} + ( -17 + \zeta_{16} + 8 \zeta_{16}^{2} - 12 \zeta_{16}^{3} + 12 \zeta_{16}^{5} - 8 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{18} + ( -4 + 13 \zeta_{16} + 10 \zeta_{16}^{2} - 10 \zeta_{16}^{4} - 13 \zeta_{16}^{5} + 4 \zeta_{16}^{6} - 6 \zeta_{16}^{7} ) q^{19} + ( -5 - 5 \zeta_{16} + 6 \zeta_{16}^{2} + 6 \zeta_{16}^{3} - 6 \zeta_{16}^{6} - 6 \zeta_{16}^{7} ) q^{20} + ( -8 + 3 \zeta_{16} - 3 \zeta_{16}^{3} + 8 \zeta_{16}^{4} + 3 \zeta_{16}^{5} + 2 \zeta_{16}^{6} + 3 \zeta_{16}^{7} ) q^{21} + ( -\zeta_{16} - 5 \zeta_{16}^{2} - 5 \zeta_{16}^{3} - \zeta_{16}^{4} - 2 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{22} + ( -1 + 2 \zeta_{16} - 13 \zeta_{16}^{2} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{4} + 13 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{23} + ( 3 - 3 \zeta_{16}^{3} - 6 \zeta_{16}^{4} - 11 \zeta_{16}^{5} - 11 \zeta_{16}^{6} - 6 \zeta_{16}^{7} ) q^{24} + ( 2 - 2 \zeta_{16}^{2} + 17 \zeta_{16}^{4} + 5 \zeta_{16}^{5} + 17 \zeta_{16}^{6} ) q^{25} + ( -12 \zeta_{16} + 7 \zeta_{16}^{3} - 5 \zeta_{16}^{4} + 5 \zeta_{16}^{6} - 7 \zeta_{16}^{7} ) q^{26} + ( 12 - 4 \zeta_{16} - 4 \zeta_{16}^{2} + 12 \zeta_{16}^{3} - 14 \zeta_{16}^{4} - 8 \zeta_{16}^{5} + 8 \zeta_{16}^{6} + 14 \zeta_{16}^{7} ) q^{27} + ( 10 - 5 \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{4} - 5 \zeta_{16}^{6} + 10 \zeta_{16}^{7} ) q^{28} + ( 3 + 18 \zeta_{16}^{2} - 18 \zeta_{16}^{3} - 3 \zeta_{16}^{5} ) q^{29} + ( 21 + 17 \zeta_{16} - 2 \zeta_{16}^{2} + 17 \zeta_{16}^{3} + 21 \zeta_{16}^{4} - 5 \zeta_{16}^{5} + 5 \zeta_{16}^{7} ) q^{30} + ( 4 - 4 \zeta_{16} - 11 \zeta_{16}^{2} - 18 \zeta_{16}^{3} + 3 \zeta_{16}^{4} + 3 \zeta_{16}^{5} - 18 \zeta_{16}^{6} - 11 \zeta_{16}^{7} ) q^{31} + ( -3 + 18 \zeta_{16} + 2 \zeta_{16}^{2} - 13 \zeta_{16}^{3} + 2 \zeta_{16}^{4} + 18 \zeta_{16}^{5} - 3 \zeta_{16}^{6} ) q^{32} + ( -\zeta_{16} + 2 \zeta_{16}^{2} + 13 \zeta_{16}^{3} + 2 \zeta_{16}^{4} + 13 \zeta_{16}^{5} + 2 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{33} + ( 8 + 2 \zeta_{16} - 12 \zeta_{16}^{2} - \zeta_{16}^{4} - 23 \zeta_{16}^{5} - 5 \zeta_{16}^{6} + 10 \zeta_{16}^{7} ) q^{34} + ( 10 + 5 \zeta_{16} + 5 \zeta_{16}^{2} + 7 \zeta_{16}^{3} - 7 \zeta_{16}^{5} - 5 \zeta_{16}^{6} - 5 \zeta_{16}^{7} ) q^{35} + ( -13 - 13 \zeta_{16} + 9 \zeta_{16}^{2} - 9 \zeta_{16}^{4} + 13 \zeta_{16}^{5} + 13 \zeta_{16}^{6} - 29 \zeta_{16}^{7} ) q^{36} + ( -21 - 21 \zeta_{16} + 3 \zeta_{16}^{2} + 6 \zeta_{16}^{3} - 7 \zeta_{16}^{4} + 7 \zeta_{16}^{5} - 6 \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{37} + ( 1 - 3 \zeta_{16} + 3 \zeta_{16}^{3} - \zeta_{16}^{4} + 9 \zeta_{16}^{5} + 26 \zeta_{16}^{6} + 9 \zeta_{16}^{7} ) q^{38} + ( -9 + 17 \zeta_{16} + 17 \zeta_{16}^{4} - 9 \zeta_{16}^{5} - 12 \zeta_{16}^{6} + 12 \zeta_{16}^{7} ) q^{39} + ( -25 - 12 \zeta_{16} + \zeta_{16}^{2} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{4} - \zeta_{16}^{5} + 12 \zeta_{16}^{6} + 25 \zeta_{16}^{7} ) q^{40} + ( 26 \zeta_{16} - 26 \zeta_{16}^{2} + 7 \zeta_{16}^{4} - 18 \zeta_{16}^{5} - 18 \zeta_{16}^{6} + 7 \zeta_{16}^{7} ) q^{41} + ( -9 + 9 \zeta_{16}^{2} - 14 \zeta_{16}^{3} - 11 \zeta_{16}^{4} + 4 \zeta_{16}^{5} - 11 \zeta_{16}^{6} - 14 \zeta_{16}^{7} ) q^{42} + ( 12 - 24 \zeta_{16} + 12 \zeta_{16}^{2} + 7 \zeta_{16}^{3} - 6 \zeta_{16}^{4} + 6 \zeta_{16}^{6} - 7 \zeta_{16}^{7} ) q^{43} + ( -12 + 7 \zeta_{16} + 7 \zeta_{16}^{2} - 12 \zeta_{16}^{3} + 5 \zeta_{16}^{4} - 14 \zeta_{16}^{5} + 14 \zeta_{16}^{6} - 5 \zeta_{16}^{7} ) q^{44} + ( -11 + 6 \zeta_{16}^{2} - 32 \zeta_{16}^{3} - 32 \zeta_{16}^{4} + 6 \zeta_{16}^{5} - 11 \zeta_{16}^{7} ) q^{45} + ( \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{4} - 10 \zeta_{16}^{6} - 10 \zeta_{16}^{7} ) q^{46} + ( -10 + 11 \zeta_{16} - 4 \zeta_{16}^{2} + 11 \zeta_{16}^{3} - 10 \zeta_{16}^{4} + 3 \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{47} + ( 11 - 11 \zeta_{16} - 5 \zeta_{16}^{2} + 32 \zeta_{16}^{3} + 2 \zeta_{16}^{4} + 2 \zeta_{16}^{5} + 32 \zeta_{16}^{6} - 5 \zeta_{16}^{7} ) q^{48} + ( 1 - 10 \zeta_{16} + 9 \zeta_{16}^{2} + 29 \zeta_{16}^{3} + 9 \zeta_{16}^{4} - 10 \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{49} + ( -3 \zeta_{16} - 19 \zeta_{16}^{2} - 17 \zeta_{16}^{3} - \zeta_{16}^{4} - 17 \zeta_{16}^{5} - 19 \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{50} + ( -22 - 7 \zeta_{16} + 29 \zeta_{16}^{2} - 15 \zeta_{16}^{3} - 3 \zeta_{16}^{4} + 40 \zeta_{16}^{5} + 14 \zeta_{16}^{6} + 8 \zeta_{16}^{7} ) q^{51} + ( 30 - 7 \zeta_{16} - 12 \zeta_{16}^{2} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + 12 \zeta_{16}^{6} + 7 \zeta_{16}^{7} ) q^{52} + ( 12 - 6 \zeta_{16} - 11 \zeta_{16}^{2} + 11 \zeta_{16}^{4} + 6 \zeta_{16}^{5} - 12 \zeta_{16}^{6} + 36 \zeta_{16}^{7} ) q^{53} + ( 26 + 26 \zeta_{16} - 32 \zeta_{16}^{2} + 4 \zeta_{16}^{3} + 24 \zeta_{16}^{4} - 24 \zeta_{16}^{5} - 4 \zeta_{16}^{6} + 32 \zeta_{16}^{7} ) q^{54} + ( -1 - 3 \zeta_{16} + 3 \zeta_{16}^{3} + \zeta_{16}^{4} - 17 \zeta_{16}^{5} - 26 \zeta_{16}^{6} - 17 \zeta_{16}^{7} ) q^{55} + ( 9 + 6 \zeta_{16} + 8 \zeta_{16}^{2} + 8 \zeta_{16}^{3} + 6 \zeta_{16}^{4} + 9 \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{56} + ( 31 + 25 \zeta_{16} + 5 \zeta_{16}^{2} - 21 \zeta_{16}^{3} + 21 \zeta_{16}^{4} - 5 \zeta_{16}^{5} - 25 \zeta_{16}^{6} - 31 \zeta_{16}^{7} ) q^{57} + ( -15 \zeta_{16} + 15 \zeta_{16}^{2} - 12 \zeta_{16}^{4} + 18 \zeta_{16}^{5} + 18 \zeta_{16}^{6} - 12 \zeta_{16}^{7} ) q^{58} + ( 1 - \zeta_{16}^{2} - 9 \zeta_{16}^{3} + 5 \zeta_{16}^{4} + 16 \zeta_{16}^{5} + 5 \zeta_{16}^{6} - 9 \zeta_{16}^{7} ) q^{59} + ( 2 + 36 \zeta_{16} + 2 \zeta_{16}^{2} - 15 \zeta_{16}^{3} + 12 \zeta_{16}^{4} - 12 \zeta_{16}^{6} + 15 \zeta_{16}^{7} ) q^{60} + ( 33 + 2 \zeta_{16} + 2 \zeta_{16}^{2} + 33 \zeta_{16}^{3} + 7 \zeta_{16}^{4} + 11 \zeta_{16}^{5} - 11 \zeta_{16}^{6} - 7 \zeta_{16}^{7} ) q^{61} + ( -17 - 3 \zeta_{16} + 32 \zeta_{16}^{2} + 12 \zeta_{16}^{3} + 12 \zeta_{16}^{4} + 32 \zeta_{16}^{5} - 3 \zeta_{16}^{6} - 17 \zeta_{16}^{7} ) q^{62} + ( 1 + 2 \zeta_{16} - 27 \zeta_{16}^{2} + 27 \zeta_{16}^{3} - 2 \zeta_{16}^{4} - \zeta_{16}^{5} + 12 \zeta_{16}^{6} + 12 \zeta_{16}^{7} ) q^{63} + ( -15 - 36 \zeta_{16} + 15 \zeta_{16}^{2} - 36 \zeta_{16}^{3} - 15 \zeta_{16}^{4} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{64} + ( -4 + 4 \zeta_{16} + 20 \zeta_{16}^{2} + 9 \zeta_{16}^{3} - 33 \zeta_{16}^{4} - 33 \zeta_{16}^{5} + 9 \zeta_{16}^{6} + 20 \zeta_{16}^{7} ) q^{65} + ( 1 - 16 \zeta_{16} - 15 \zeta_{16}^{2} - 15 \zeta_{16}^{4} - 16 \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{66} + ( 3 \zeta_{16} + 8 \zeta_{16}^{2} - 25 \zeta_{16}^{3} + 4 \zeta_{16}^{4} - 25 \zeta_{16}^{5} + 8 \zeta_{16}^{6} + 3 \zeta_{16}^{7} ) q^{67} + ( -22 + 17 \zeta_{16} - 35 \zeta_{16}^{2} + 9 \zeta_{16}^{3} + 15 \zeta_{16}^{4} + 3 \zeta_{16}^{5} - 29 \zeta_{16}^{6} - 5 \zeta_{16}^{7} ) q^{68} + ( -26 - 5 \zeta_{16} - 14 \zeta_{16}^{2} + 11 \zeta_{16}^{3} - 11 \zeta_{16}^{5} + 14 \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{69} + ( -10 - 3 \zeta_{16} - 12 \zeta_{16}^{2} + 12 \zeta_{16}^{4} + 3 \zeta_{16}^{5} + 10 \zeta_{16}^{6} + 24 \zeta_{16}^{7} ) q^{70} + ( 4 + 4 \zeta_{16} + 6 \zeta_{16}^{2} - 41 \zeta_{16}^{3} - 17 \zeta_{16}^{4} + 17 \zeta_{16}^{5} + 41 \zeta_{16}^{6} - 6 \zeta_{16}^{7} ) q^{71} + ( 3 - 3 \zeta_{16} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{4} - 22 \zeta_{16}^{5} - 21 \zeta_{16}^{6} - 22 \zeta_{16}^{7} ) q^{72} + ( 3 - 30 \zeta_{16} - 28 \zeta_{16}^{2} - 28 \zeta_{16}^{3} - 30 \zeta_{16}^{4} + 3 \zeta_{16}^{5} + 20 \zeta_{16}^{6} - 20 \zeta_{16}^{7} ) q^{73} + ( 22 - 13 \zeta_{16} + 21 \zeta_{16}^{2} + 31 \zeta_{16}^{3} - 31 \zeta_{16}^{4} - 21 \zeta_{16}^{5} + 13 \zeta_{16}^{6} - 22 \zeta_{16}^{7} ) q^{74} + ( -24 - 33 \zeta_{16} + 33 \zeta_{16}^{2} + 24 \zeta_{16}^{3} + 21 \zeta_{16}^{4} + 34 \zeta_{16}^{5} + 34 \zeta_{16}^{6} + 21 \zeta_{16}^{7} ) q^{75} + ( -10 + 10 \zeta_{16}^{2} + 47 \zeta_{16}^{3} - 24 \zeta_{16}^{4} - 8 \zeta_{16}^{5} - 24 \zeta_{16}^{6} + 47 \zeta_{16}^{7} ) q^{76} + ( -27 + 16 \zeta_{16} - 27 \zeta_{16}^{2} + 12 \zeta_{16}^{3} - 15 \zeta_{16}^{4} + 15 \zeta_{16}^{6} - 12 \zeta_{16}^{7} ) q^{77} + ( -46 + 21 \zeta_{16} + 21 \zeta_{16}^{2} - 46 \zeta_{16}^{3} + 29 \zeta_{16}^{5} - 29 \zeta_{16}^{6} ) q^{78} + ( -12 + 17 \zeta_{16} - 44 \zeta_{16}^{2} + 41 \zeta_{16}^{3} + 41 \zeta_{16}^{4} - 44 \zeta_{16}^{5} + 17 \zeta_{16}^{6} - 12 \zeta_{16}^{7} ) q^{79} + ( 3 + \zeta_{16} + 10 \zeta_{16}^{2} - 10 \zeta_{16}^{3} - \zeta_{16}^{4} - 3 \zeta_{16}^{5} - 41 \zeta_{16}^{6} - 41 \zeta_{16}^{7} ) q^{80} + ( -28 - 35 \zeta_{16} + 9 \zeta_{16}^{2} - 35 \zeta_{16}^{3} - 28 \zeta_{16}^{4} + 3 \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{81} + ( -51 + 51 \zeta_{16} + 18 \zeta_{16}^{2} - 40 \zeta_{16}^{3} + 44 \zeta_{16}^{4} + 44 \zeta_{16}^{5} - 40 \zeta_{16}^{6} + 18 \zeta_{16}^{7} ) q^{82} + ( -11 + 11 \zeta_{16} + 63 \zeta_{16}^{2} - 10 \zeta_{16}^{3} + 63 \zeta_{16}^{4} + 11 \zeta_{16}^{5} - 11 \zeta_{16}^{6} ) q^{83} + ( -15 \zeta_{16} + 2 \zeta_{16}^{2} + 13 \zeta_{16}^{3} - 30 \zeta_{16}^{4} + 13 \zeta_{16}^{5} + 2 \zeta_{16}^{6} - 15 \zeta_{16}^{7} ) q^{84} + ( 64 + 62 \zeta_{16} - 6 \zeta_{16}^{2} - 18 \zeta_{16}^{3} + 3 \zeta_{16}^{4} - \zeta_{16}^{5} - 6 \zeta_{16}^{6} - 18 \zeta_{16}^{7} ) q^{85} + ( 36 - 19 \zeta_{16} - 25 \zeta_{16}^{2} + 37 \zeta_{16}^{3} - 37 \zeta_{16}^{5} + 25 \zeta_{16}^{6} + 19 \zeta_{16}^{7} ) q^{86} + ( 39 + 15 \zeta_{16} - 27 \zeta_{16}^{2} + 27 \zeta_{16}^{4} - 15 \zeta_{16}^{5} - 39 \zeta_{16}^{6} + 24 \zeta_{16}^{7} ) q^{87} + ( 22 + 22 \zeta_{16} + 14 \zeta_{16}^{2} - 7 \zeta_{16}^{3} + 3 \zeta_{16}^{4} - 3 \zeta_{16}^{5} + 7 \zeta_{16}^{6} - 14 \zeta_{16}^{7} ) q^{88} + ( 36 - 8 \zeta_{16} + 8 \zeta_{16}^{3} - 36 \zeta_{16}^{4} + 47 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + 47 \zeta_{16}^{7} ) q^{89} + ( 32 - 23 \zeta_{16} + 43 \zeta_{16}^{2} + 43 \zeta_{16}^{3} - 23 \zeta_{16}^{4} + 32 \zeta_{16}^{5} + 49 \zeta_{16}^{6} - 49 \zeta_{16}^{7} ) q^{90} + ( -3 - 2 \zeta_{16} + 4 \zeta_{16}^{2} - 23 \zeta_{16}^{3} + 23 \zeta_{16}^{4} - 4 \zeta_{16}^{5} + 2 \zeta_{16}^{6} + 3 \zeta_{16}^{7} ) q^{91} + ( 42 - 25 \zeta_{16} + 25 \zeta_{16}^{2} - 42 \zeta_{16}^{3} + 9 \zeta_{16}^{4} + 6 \zeta_{16}^{5} + 6 \zeta_{16}^{6} + 9 \zeta_{16}^{7} ) q^{92} + ( 35 - 35 \zeta_{16}^{2} + 12 \zeta_{16}^{3} - 31 \zeta_{16}^{4} - 86 \zeta_{16}^{5} - 31 \zeta_{16}^{6} + 12 \zeta_{16}^{7} ) q^{93} + ( -1 - 2 \zeta_{16} - \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 9 \zeta_{16}^{4} + 9 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{94} + ( -19 - 100 \zeta_{16} - 100 \zeta_{16}^{2} - 19 \zeta_{16}^{3} + 27 \zeta_{16}^{4} + 18 \zeta_{16}^{5} - 18 \zeta_{16}^{6} - 27 \zeta_{16}^{7} ) q^{95} + ( 41 + 10 \zeta_{16} - 51 \zeta_{16}^{2} + 58 \zeta_{16}^{3} + 58 \zeta_{16}^{4} - 51 \zeta_{16}^{5} + 10 \zeta_{16}^{6} + 41 \zeta_{16}^{7} ) q^{96} + ( -43 + 35 \zeta_{16} - 26 \zeta_{16}^{2} + 26 \zeta_{16}^{3} - 35 \zeta_{16}^{4} + 43 \zeta_{16}^{5} + 63 \zeta_{16}^{6} + 63 \zeta_{16}^{7} ) q^{97} + ( 2 + \zeta_{16} - 31 \zeta_{16}^{2} + \zeta_{16}^{3} + 2 \zeta_{16}^{4} - 40 \zeta_{16}^{5} + 40 \zeta_{16}^{7} ) q^{98} + ( 17 - 17 \zeta_{16} + 18 \zeta_{16}^{2} - 15 \zeta_{16}^{3} + 32 \zeta_{16}^{4} + 32 \zeta_{16}^{5} - 15 \zeta_{16}^{6} + 18 \zeta_{16}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} - 24q^{5} - 16q^{7} + 16q^{8} + 8q^{9} + O(q^{10})$$ $$8q - 8q^{4} - 24q^{5} - 16q^{7} + 16q^{8} + 8q^{9} + 40q^{11} + 40q^{12} + 16q^{14} + 32q^{15} - 16q^{17} - 136q^{18} - 32q^{19} - 40q^{20} - 64q^{21} - 8q^{23} + 24q^{24} + 16q^{25} + 96q^{27} + 80q^{28} + 24q^{29} + 168q^{30} + 32q^{31} - 24q^{32} + 64q^{34} + 80q^{35} - 104q^{36} - 168q^{37} + 8q^{38} - 72q^{39} - 200q^{40} - 72q^{42} + 96q^{43} - 96q^{44} - 88q^{45} - 80q^{47} + 88q^{48} + 8q^{49} - 176q^{51} + 240q^{52} + 96q^{53} + 208q^{54} - 8q^{55} + 72q^{56} + 248q^{57} + 8q^{59} + 16q^{60} + 264q^{61} - 136q^{62} + 8q^{63} - 120q^{64} - 32q^{65} + 8q^{66} - 176q^{68} - 208q^{69} - 80q^{70} + 32q^{71} + 24q^{72} + 24q^{73} + 176q^{74} - 192q^{75} - 80q^{76} - 216q^{77} - 368q^{78} - 96q^{79} + 24q^{80} - 224q^{81} - 408q^{82} - 88q^{83} + 512q^{85} + 288q^{86} + 312q^{87} + 176q^{88} + 288q^{89} + 256q^{90} - 24q^{91} + 336q^{92} + 280q^{93} - 8q^{94} - 152q^{95} + 328q^{96} - 344q^{97} + 16q^{98} + 136q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/17\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$\zeta_{16}$$

## 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
 0.923880 + 0.382683i −0.382683 + 0.923880i 0.923880 − 0.382683i −0.382683 − 0.923880i 0.382683 + 0.923880i −0.923880 + 0.382683i 0.382683 − 0.923880i −0.923880 − 0.382683i
−1.63099 + 0.675577i 3.96908 + 0.789499i −0.624715 + 0.624715i −4.29916 6.43416i −7.00688 + 1.39376i −2.27356 + 3.40262i 3.29916 7.96489i 6.81537 + 2.82302i 11.3586 + 7.58960i
5.1 1.08979 + 2.63099i −2.88669 4.32023i −2.90602 + 2.90602i −0.711297 + 3.57593i 8.22059 12.3030i −0.644047 3.23784i −0.288703 0.119585i −6.88730 + 16.6274i −10.1834 + 2.02560i
6.1 −1.63099 0.675577i 3.96908 0.789499i −0.624715 0.624715i −4.29916 + 6.43416i −7.00688 1.39376i −2.27356 3.40262i 3.29916 + 7.96489i 6.81537 2.82302i 11.3586 7.58960i
7.1 1.08979 2.63099i −2.88669 + 4.32023i −2.90602 2.90602i −0.711297 3.57593i 8.22059 + 12.3030i −0.644047 + 3.23784i −0.288703 + 0.119585i −6.88730 16.6274i −10.1834 2.02560i
10.1 0.324423 0.783227i −1.35595 0.906019i 2.32023 + 2.32023i −6.70292 + 1.33329i −1.14952 + 0.768086i 0.886687 + 0.176373i 5.70292 2.36223i −2.42641 5.85788i −1.13031 + 5.68246i
11.1 0.216773 + 0.0897902i 0.273561 + 1.37529i −2.78950 2.78950i −0.286621 0.191514i −0.0641865 + 0.322688i −5.96908 + 3.98841i −0.713379 1.72225i 6.49834 2.69170i −0.0449356 0.0672509i
12.1 0.324423 + 0.783227i −1.35595 + 0.906019i 2.32023 2.32023i −6.70292 1.33329i −1.14952 0.768086i 0.886687 0.176373i 5.70292 + 2.36223i −2.42641 + 5.85788i −1.13031 5.68246i
14.1 0.216773 0.0897902i 0.273561 1.37529i −2.78950 + 2.78950i −0.286621 + 0.191514i −0.0641865 0.322688i −5.96908 3.98841i −0.713379 + 1.72225i 6.49834 + 2.69170i −0.0449356 + 0.0672509i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 14.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.3.e.b 8
3.b odd 2 1 153.3.p.a 8
4.b odd 2 1 272.3.bh.b 8
5.b even 2 1 425.3.u.a 8
5.c odd 4 1 425.3.t.b 8
5.c odd 4 1 425.3.t.d 8
17.b even 2 1 289.3.e.g 8
17.c even 4 1 289.3.e.f 8
17.c even 4 1 289.3.e.h 8
17.d even 8 1 289.3.e.a 8
17.d even 8 1 289.3.e.e 8
17.d even 8 1 289.3.e.j 8
17.d even 8 1 289.3.e.n 8
17.e odd 16 1 inner 17.3.e.b 8
17.e odd 16 1 289.3.e.a 8
17.e odd 16 1 289.3.e.e 8
17.e odd 16 1 289.3.e.f 8
17.e odd 16 1 289.3.e.g 8
17.e odd 16 1 289.3.e.h 8
17.e odd 16 1 289.3.e.j 8
17.e odd 16 1 289.3.e.n 8
51.i even 16 1 153.3.p.a 8
68.i even 16 1 272.3.bh.b 8
85.o even 16 1 425.3.t.d 8
85.p odd 16 1 425.3.u.a 8
85.r even 16 1 425.3.t.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.3.e.b 8 1.a even 1 1 trivial
17.3.e.b 8 17.e odd 16 1 inner
153.3.p.a 8 3.b odd 2 1
153.3.p.a 8 51.i even 16 1
272.3.bh.b 8 4.b odd 2 1
272.3.bh.b 8 68.i even 16 1
289.3.e.a 8 17.d even 8 1
289.3.e.a 8 17.e odd 16 1
289.3.e.e 8 17.d even 8 1
289.3.e.e 8 17.e odd 16 1
289.3.e.f 8 17.c even 4 1
289.3.e.f 8 17.e odd 16 1
289.3.e.g 8 17.b even 2 1
289.3.e.g 8 17.e odd 16 1
289.3.e.h 8 17.c even 4 1
289.3.e.h 8 17.e odd 16 1
289.3.e.j 8 17.d even 8 1
289.3.e.j 8 17.e odd 16 1
289.3.e.n 8 17.d even 8 1
289.3.e.n 8 17.e odd 16 1
425.3.t.b 8 5.c odd 4 1
425.3.t.b 8 85.r even 16 1
425.3.t.d 8 5.c odd 4 1
425.3.t.d 8 85.o even 16 1
425.3.u.a 8 5.b even 2 1
425.3.u.a 8 85.p odd 16 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 4 T_{2}^{6} + 16 T_{2}^{5} + 8 T_{2}^{4} - 8 T_{2}^{3} + 20 T_{2}^{2} - 8 T_{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(17, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T^{2} - 16 T^{3} + 8 T^{4} - 72 T^{5} + 148 T^{6} - 200 T^{7} + 801 T^{8} - 800 T^{9} + 2368 T^{10} - 4608 T^{11} + 2048 T^{12} - 16384 T^{13} + 16384 T^{14} + 65536 T^{16}$$
$3$ $$1 - 4 T^{2} + 16 T^{3} + 28 T^{4} - 376 T^{5} + 444 T^{6} + 8 T^{7} - 6760 T^{8} + 72 T^{9} + 35964 T^{10} - 274104 T^{11} + 183708 T^{12} + 944784 T^{13} - 2125764 T^{14} + 43046721 T^{16}$$
$5$ $$1 + 24 T + 280 T^{2} + 2168 T^{3} + 13330 T^{4} + 75080 T^{5} + 410024 T^{6} + 2141288 T^{7} + 10795618 T^{8} + 53532200 T^{9} + 256265000 T^{10} + 1173125000 T^{11} + 5207031250 T^{12} + 21171875000 T^{13} + 68359375000 T^{14} + 146484375000 T^{15} + 152587890625 T^{16}$$
$7$ $$1 + 16 T + 124 T^{2} + 1296 T^{3} + 10140 T^{4} + 48232 T^{5} + 458172 T^{6} + 3313432 T^{7} + 15111448 T^{8} + 162358168 T^{9} + 1100070972 T^{10} + 5674446568 T^{11} + 58455082140 T^{12} + 366087922704 T^{13} + 1716319612924 T^{14} + 10851569165584 T^{15} + 33232930569601 T^{16}$$
$11$ $$1 - 40 T + 700 T^{2} - 7600 T^{3} + 87412 T^{4} - 1438024 T^{5} + 19499716 T^{6} - 176624752 T^{7} + 1537072952 T^{8} - 21371594992 T^{9} + 285495341956 T^{10} - 2547547235464 T^{11} + 18737538505972 T^{12} - 197124426967600 T^{13} + 2196899863704700 T^{14} - 15189993343329640 T^{15} + 45949729863572161 T^{16}$$
$13$ $$1 + 784 T^{3} + 9888 T^{4} - 237552 T^{5} + 307328 T^{6} - 20917120 T^{7} + 236770498 T^{8} - 3534993280 T^{9} + 8777595008 T^{10} - 1146618131568 T^{11} + 8065945369248 T^{12} + 108081057609616 T^{13} + 665416609183179841 T^{16}$$
$17$ $$1 + 16 T + 240 T^{2} - 784 T^{3} - 11934 T^{4} - 226576 T^{5} + 20045040 T^{6} + 386201104 T^{7} + 6975757441 T^{8}$$
$19$ $$1 + 32 T + 544 T^{2} + 10608 T^{3} + 192512 T^{4} + 3096432 T^{5} + 96494752 T^{6} + 2593739296 T^{7} + 50479240962 T^{8} + 936339885856 T^{9} + 12575292575392 T^{10} + 145674371396592 T^{11} + 3269539688148992 T^{12} + 65038350862753008 T^{13} + 1204043315971991584 T^{14} + 25568213945052291872 T^{15} +$$$$28\!\cdots\!81$$$$T^{16}$$
$23$ $$1 + 8 T + 732 T^{2} + 3584 T^{3} + 513652 T^{4} + 7272312 T^{5} + 445470244 T^{6} + 2689381328 T^{7} + 220561130936 T^{8} + 1422682722512 T^{9} + 124660838551204 T^{10} + 1076563172005368 T^{11} + 40224594211556212 T^{12} + 148472616189718016 T^{13} + 16041505084238874972 T^{14} + 92742690596309998472 T^{15} +$$$$61\!\cdots\!61$$$$T^{16}$$
$29$ $$1 - 24 T + 684 T^{2} - 83736 T^{3} + 1812294 T^{4} - 39076344 T^{5} + 2881128204 T^{6} - 72099991512 T^{7} + 1161408147714 T^{8} - 60636092861592 T^{9} + 2037767237253324 T^{10} - 23243520710618424 T^{11} + 906593572730742534 T^{12} - 35228340887625630936 T^{13} +$$$$24\!\cdots\!44$$$$T^{14} -$$$$71\!\cdots\!44$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16}$$
$31$ $$1 - 32 T + 1988 T^{2} - 84680 T^{3} + 2224116 T^{4} - 107632640 T^{5} + 3131219084 T^{6} - 96339571608 T^{7} + 3769094500984 T^{8} - 92582328315288 T^{9} + 2891746579674764 T^{10} - 95524364195747840 T^{11} + 1896928602629127156 T^{12} - 69406123341534228680 T^{13} +$$$$15\!\cdots\!68$$$$T^{14} -$$$$24\!\cdots\!72$$$$T^{15} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$1 + 168 T + 13240 T^{2} + 650184 T^{3} + 19962450 T^{4} + 154153272 T^{5} - 24329640440 T^{6} - 1835917350504 T^{7} - 81344425680670 T^{8} - 2513370852839976 T^{9} - 45597663256670840 T^{10} + 395515121004160248 T^{11} + 70117695474925266450 T^{12} +$$$$31\!\cdots\!16$$$$T^{13} +$$$$87\!\cdots\!40$$$$T^{14} +$$$$15\!\cdots\!52$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16}$$
$41$ $$1 + 5396 T^{2} - 54048 T^{3} + 11503750 T^{4} - 349337248 T^{5} + 11972012020 T^{6} - 1013376724544 T^{7} + 10882038609154 T^{8} - 1703486273958464 T^{9} + 33830044657647220 T^{10} - 1659388343264068768 T^{11} + 91856583604500703750 T^{12} -$$$$72\!\cdots\!48$$$$T^{13} +$$$$12\!\cdots\!76$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16}$$
$43$ $$1 - 96 T + 6096 T^{2} - 290832 T^{3} + 11553408 T^{4} - 238904592 T^{5} - 4015115952 T^{6} + 725328019488 T^{7} - 38391385271230 T^{8} + 1341131508033312 T^{9} - 13726882431813552 T^{10} - 1510202660105221008 T^{11} +$$$$13\!\cdots\!08$$$$T^{12} -$$$$62\!\cdots\!68$$$$T^{13} +$$$$24\!\cdots\!96$$$$T^{14} -$$$$70\!\cdots\!04$$$$T^{15} +$$$$13\!\cdots\!01$$$$T^{16}$$
$47$ $$1 + 80 T + 3200 T^{2} + 228944 T^{3} + 25698900 T^{4} + 1284226160 T^{5} + 46709290368 T^{6} + 3079489449072 T^{7} + 201735188722534 T^{8} + 6802592193000048 T^{9} + 227926436732212608 T^{10} + 13842950309774806640 T^{11} +$$$$61\!\cdots\!00$$$$T^{12} +$$$$12\!\cdots\!56$$$$T^{13} +$$$$37\!\cdots\!00$$$$T^{14} +$$$$20\!\cdots\!20$$$$T^{15} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$1 - 96 T + 3844 T^{2} - 250080 T^{3} + 17203208 T^{4} - 878356128 T^{5} + 61303275468 T^{6} - 2343880946208 T^{7} + 44401903321166 T^{8} - 6583961577898272 T^{9} + 483712330318020108 T^{10} - 19468202420862148512 T^{11} +$$$$10\!\cdots\!88$$$$T^{12} -$$$$43\!\cdots\!20$$$$T^{13} +$$$$18\!\cdots\!04$$$$T^{14} -$$$$13\!\cdots\!24$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16}$$
$59$ $$1 - 8 T - 1064 T^{2} + 139576 T^{3} - 481440 T^{4} + 260011896 T^{5} + 38349218008 T^{6} - 28053020936 T^{7} + 74913788078050 T^{8} - 97652565878216 T^{9} + 464691318670636888 T^{10} + 10967440526288193336 T^{11} - 70690045880224302240 T^{12} +$$$$71\!\cdots\!76$$$$T^{13} -$$$$18\!\cdots\!84$$$$T^{14} -$$$$49\!\cdots\!88$$$$T^{15} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$1 - 264 T + 33304 T^{2} - 2900248 T^{3} + 200549746 T^{4} - 11914556776 T^{5} + 710898258152 T^{6} - 45325216747576 T^{7} + 2838106764821794 T^{8} - 168655131517730296 T^{9} + 9842984249549545832 T^{10} -$$$$61\!\cdots\!36$$$$T^{11} +$$$$38\!\cdots\!26$$$$T^{12} -$$$$20\!\cdots\!48$$$$T^{13} +$$$$88\!\cdots\!84$$$$T^{14} -$$$$26\!\cdots\!24$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16}$$
$67$ $$1 - 30264 T^{2} + 421009100 T^{4} - 3513162425416 T^{6} + 19237575139662822 T^{8} - 70794161127211291336 T^{10} +$$$$17\!\cdots\!00$$$$T^{12} -$$$$24\!\cdots\!04$$$$T^{14} +$$$$16\!\cdots\!81$$$$T^{16}$$
$71$ $$1 - 32 T - 2196 T^{2} + 376992 T^{3} - 40999012 T^{4} + 1392153480 T^{5} + 50862865468 T^{6} - 10985693543800 T^{7} + 1117214885242200 T^{8} - 55378881154295800 T^{9} + 1292510912018731708 T^{10} +$$$$17\!\cdots\!80$$$$T^{11} -$$$$26\!\cdots\!32$$$$T^{12} +$$$$12\!\cdots\!92$$$$T^{13} -$$$$36\!\cdots\!36$$$$T^{14} -$$$$26\!\cdots\!92$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16}$$
$73$ $$1 - 24 T + 3500 T^{2} + 527224 T^{3} - 252666 T^{4} - 799621288 T^{5} + 103501902988 T^{6} + 4216386556104 T^{7} - 1450386823086462 T^{8} + 22469123957478216 T^{9} + 2939271985011844108 T^{10} -$$$$12\!\cdots\!32$$$$T^{11} -$$$$20\!\cdots\!46$$$$T^{12} +$$$$22\!\cdots\!76$$$$T^{13} +$$$$80\!\cdots\!00$$$$T^{14} -$$$$29\!\cdots\!16$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16}$$
$79$ $$1 + 96 T - 11292 T^{2} - 1308856 T^{3} + 36577844 T^{4} + 2295987344 T^{5} - 542602414484 T^{6} + 12994739703176 T^{7} + 6454474657365240 T^{8} + 81100170487521416 T^{9} - 21134407994947373204 T^{10} +$$$$55\!\cdots\!24$$$$T^{11} +$$$$55\!\cdots\!84$$$$T^{12} -$$$$12\!\cdots\!56$$$$T^{13} -$$$$66\!\cdots\!72$$$$T^{14} +$$$$35\!\cdots\!76$$$$T^{15} +$$$$23\!\cdots\!21$$$$T^{16}$$
$83$ $$1 + 88 T + 12840 T^{2} - 772808 T^{3} - 70014880 T^{4} - 15998431560 T^{5} + 276363214376 T^{6} + 33752671318936 T^{7} + 12884842311213474 T^{8} + 232522152716150104 T^{9} + 13115734140448022696 T^{10} -$$$$52\!\cdots\!40$$$$T^{11} -$$$$15\!\cdots\!80$$$$T^{12} -$$$$11\!\cdots\!92$$$$T^{13} +$$$$13\!\cdots\!40$$$$T^{14} +$$$$64\!\cdots\!52$$$$T^{15} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$1 - 288 T + 41472 T^{2} - 4656976 T^{3} + 519733472 T^{4} - 56560463824 T^{5} + 5578739762816 T^{6} - 499331992822432 T^{7} + 43753371662175234 T^{8} - 3955208715146483872 T^{9} +$$$$35\!\cdots\!56$$$$T^{10} -$$$$28\!\cdots\!64$$$$T^{11} +$$$$20\!\cdots\!32$$$$T^{12} -$$$$14\!\cdots\!76$$$$T^{13} +$$$$10\!\cdots\!12$$$$T^{14} -$$$$56\!\cdots\!08$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16}$$
$97$ $$1 + 344 T + 70152 T^{2} + 11174248 T^{3} + 1407887122 T^{4} + 151612370328 T^{5} + 14250743174520 T^{6} + 1250314851633384 T^{7} + 116834146569341026 T^{8} + 11764212439018510056 T^{9} +$$$$12\!\cdots\!20$$$$T^{10} +$$$$12\!\cdots\!12$$$$T^{11} +$$$$11\!\cdots\!42$$$$T^{12} +$$$$82\!\cdots\!52$$$$T^{13} +$$$$48\!\cdots\!32$$$$T^{14} +$$$$22\!\cdots\!36$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16}$$