Properties

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

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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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{2} + ( -1 - \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{3} + ( -2 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 3 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{4} + ( 2 + 2 \zeta_{16} - 2 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{4} + 2 \zeta_{16}^{5} ) q^{5} + ( -1 + \zeta_{16} - 4 \zeta_{16}^{2} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{4} + 4 \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{6} + ( 1 - \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{4} - 5 \zeta_{16}^{5} - 5 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{7} + ( -3 + 3 \zeta_{16}^{2} - 3 \zeta_{16}^{3} + 5 \zeta_{16}^{4} + \zeta_{16}^{5} + 5 \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{8} + ( -2 - 3 \zeta_{16} - 2 \zeta_{16}^{2} + 4 \zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{2} + ( -1 - \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{3} + ( -2 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 3 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{4} + ( 2 + 2 \zeta_{16} - 2 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{4} + 2 \zeta_{16}^{5} ) q^{5} + ( -1 + \zeta_{16} - 4 \zeta_{16}^{2} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{4} + 4 \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{6} + ( 1 - \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{4} - 5 \zeta_{16}^{5} - 5 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{7} + ( -3 + 3 \zeta_{16}^{2} - 3 \zeta_{16}^{3} + 5 \zeta_{16}^{4} + \zeta_{16}^{5} + 5 \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{8} + ( -2 - 3 \zeta_{16} - 2 \zeta_{16}^{2} + 4 \zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{9} + ( 2 + 4 \zeta_{16} + 4 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 6 \zeta_{16}^{4} - 6 \zeta_{16}^{5} + 6 \zeta_{16}^{6} + 6 \zeta_{16}^{7} ) q^{10} + ( -1 - \zeta_{16} + 3 \zeta_{16}^{2} - 6 \zeta_{16}^{3} - 6 \zeta_{16}^{4} + 3 \zeta_{16}^{5} - \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{11} + ( 6 - 5 \zeta_{16} + 9 \zeta_{16}^{2} - 9 \zeta_{16}^{3} + 5 \zeta_{16}^{4} - 6 \zeta_{16}^{5} + \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{12} + ( 2 + 10 \zeta_{16} + 10 \zeta_{16}^{3} + 2 \zeta_{16}^{4} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{13} + ( 1 - \zeta_{16} - 13 \zeta_{16}^{2} + 3 \zeta_{16}^{3} - \zeta_{16}^{4} - \zeta_{16}^{5} + 3 \zeta_{16}^{6} - 13 \zeta_{16}^{7} ) q^{14} + ( -2 - 2 \zeta_{16} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{6} ) q^{15} + ( 4 \zeta_{16} - 8 \zeta_{16}^{2} + 4 \zeta_{16}^{3} + 3 \zeta_{16}^{4} + 4 \zeta_{16}^{5} - 8 \zeta_{16}^{6} + 4 \zeta_{16}^{7} ) q^{16} + 17 \zeta_{16}^{7} q^{17} + ( 7 - 2 \zeta_{16} + 3 \zeta_{16}^{2} - 13 \zeta_{16}^{3} + 13 \zeta_{16}^{5} - 3 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{18} + ( -7 \zeta_{16} + 6 \zeta_{16}^{2} - 6 \zeta_{16}^{4} + 7 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{19} + ( -10 - 10 \zeta_{16} + 2 \zeta_{16}^{2} + 10 \zeta_{16}^{3} + 4 \zeta_{16}^{4} - 4 \zeta_{16}^{5} - 10 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{20} + ( -8 + 2 \zeta_{16} - 2 \zeta_{16}^{3} + 8 \zeta_{16}^{4} + 14 \zeta_{16}^{6} ) q^{21} + ( -13 + 7 \zeta_{16} - 3 \zeta_{16}^{2} - 3 \zeta_{16}^{3} + 7 \zeta_{16}^{4} - 13 \zeta_{16}^{5} - 6 \zeta_{16}^{6} + 6 \zeta_{16}^{7} ) q^{22} + ( -7 - 7 \zeta_{16} - 3 \zeta_{16}^{2} + 5 \zeta_{16}^{3} - 5 \zeta_{16}^{4} + 3 \zeta_{16}^{5} + 7 \zeta_{16}^{6} + 7 \zeta_{16}^{7} ) q^{23} + ( -10 + 15 \zeta_{16} - 15 \zeta_{16}^{2} + 10 \zeta_{16}^{3} - 13 \zeta_{16}^{4} - 3 \zeta_{16}^{5} - 3 \zeta_{16}^{6} - 13 \zeta_{16}^{7} ) q^{24} + ( 8 - 8 \zeta_{16}^{2} - 16 \zeta_{16}^{3} + 4 \zeta_{16}^{4} - \zeta_{16}^{5} + 4 \zeta_{16}^{6} - 16 \zeta_{16}^{7} ) q^{25} + ( 22 - 12 \zeta_{16} + 22 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 6 \zeta_{16}^{4} - 6 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{26} + ( 5 + 10 \zeta_{16} + 10 \zeta_{16}^{2} + 5 \zeta_{16}^{3} + 2 \zeta_{16}^{4} - 3 \zeta_{16}^{5} + 3 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{27} + ( 19 + \zeta_{16} + 15 \zeta_{16}^{2} - 7 \zeta_{16}^{3} - 7 \zeta_{16}^{4} + 15 \zeta_{16}^{5} + \zeta_{16}^{6} + 19 \zeta_{16}^{7} ) q^{28} + ( 6 + 8 \zeta_{16} - 16 \zeta_{16}^{2} + 16 \zeta_{16}^{3} - 8 \zeta_{16}^{4} - 6 \zeta_{16}^{5} - 2 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{29} + ( 2 - 2 \zeta_{16} - 8 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{4} + 4 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{30} + ( 3 - 3 \zeta_{16} + 5 \zeta_{16}^{2} - 5 \zeta_{16}^{3} - 5 \zeta_{16}^{4} - 5 \zeta_{16}^{5} - 5 \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{31} + ( 11 - \zeta_{16} - 11 \zeta_{16}^{2} - 17 \zeta_{16}^{3} - 11 \zeta_{16}^{4} - \zeta_{16}^{5} + 11 \zeta_{16}^{6} ) q^{32} + ( -2 \zeta_{16} + 7 \zeta_{16}^{2} + \zeta_{16}^{3} + 12 \zeta_{16}^{4} + \zeta_{16}^{5} + 7 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{33} + ( -17 - 17 \zeta_{16} + 17 \zeta_{16}^{3} + 17 \zeta_{16}^{4} - 17 \zeta_{16}^{5} + 17 \zeta_{16}^{6} - 17 \zeta_{16}^{7} ) q^{34} + ( -20 + 20 \zeta_{16}^{2} + 8 \zeta_{16}^{3} - 8 \zeta_{16}^{5} - 20 \zeta_{16}^{6} ) q^{35} + ( -16 + 20 \zeta_{16} + 3 \zeta_{16}^{2} - 3 \zeta_{16}^{4} - 20 \zeta_{16}^{5} + 16 \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{36} + ( 4 + 4 \zeta_{16} + 12 \zeta_{16}^{2} - 2 \zeta_{16}^{3} - 16 \zeta_{16}^{4} + 16 \zeta_{16}^{5} + 2 \zeta_{16}^{6} - 12 \zeta_{16}^{7} ) q^{37} + ( -15 + 18 \zeta_{16} - 18 \zeta_{16}^{3} + 15 \zeta_{16}^{4} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{38} + ( 6 - 22 \zeta_{16} - 12 \zeta_{16}^{2} - 12 \zeta_{16}^{3} - 22 \zeta_{16}^{4} + 6 \zeta_{16}^{5} - 20 \zeta_{16}^{6} + 20 \zeta_{16}^{7} ) q^{39} + ( 8 - 14 \zeta_{16} - 6 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{4} + 6 \zeta_{16}^{5} + 14 \zeta_{16}^{6} - 8 \zeta_{16}^{7} ) q^{40} + ( 6 + 5 \zeta_{16} - 5 \zeta_{16}^{2} - 6 \zeta_{16}^{3} + 14 \zeta_{16}^{4} + 6 \zeta_{16}^{5} + 6 \zeta_{16}^{6} + 14 \zeta_{16}^{7} ) q^{41} + ( 2 - 2 \zeta_{16}^{2} + 26 \zeta_{16}^{3} - 16 \zeta_{16}^{4} + 26 \zeta_{16}^{5} - 16 \zeta_{16}^{6} + 26 \zeta_{16}^{7} ) q^{42} + ( -29 + 22 \zeta_{16} - 29 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 14 \zeta_{16}^{4} - 14 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{43} + ( 15 - 8 \zeta_{16} - 8 \zeta_{16}^{2} + 15 \zeta_{16}^{3} + 3 \zeta_{16}^{4} + 15 \zeta_{16}^{5} - 15 \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{44} + ( -14 \zeta_{16} - 16 \zeta_{16}^{2} + 20 \zeta_{16}^{3} + 20 \zeta_{16}^{4} - 16 \zeta_{16}^{5} - 14 \zeta_{16}^{6} ) q^{45} + ( -11 - 17 \zeta_{16} + 9 \zeta_{16}^{2} - 9 \zeta_{16}^{3} + 17 \zeta_{16}^{4} + 11 \zeta_{16}^{5} + \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{46} + ( 24 - 8 \zeta_{16} + 2 \zeta_{16}^{2} - 8 \zeta_{16}^{3} + 24 \zeta_{16}^{4} + 22 \zeta_{16}^{5} - 22 \zeta_{16}^{7} ) q^{47} + ( 17 - 17 \zeta_{16} + 17 \zeta_{16}^{2} - 12 \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{5} - 12 \zeta_{16}^{6} + 17 \zeta_{16}^{7} ) q^{48} + ( 2 - 4 \zeta_{16} - 10 \zeta_{16}^{2} - 7 \zeta_{16}^{3} - 10 \zeta_{16}^{4} - 4 \zeta_{16}^{5} + 2 \zeta_{16}^{6} ) q^{49} + ( 35 \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - 57 \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{6} + 35 \zeta_{16}^{7} ) q^{50} + ( 17 + 17 \zeta_{16}^{2} - 17 \zeta_{16}^{3} + 17 \zeta_{16}^{4} - 17 \zeta_{16}^{5} - 17 \zeta_{16}^{7} ) q^{51} + ( -48 + 22 \zeta_{16} - 34 \zeta_{16}^{2} + 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} + 34 \zeta_{16}^{6} - 22 \zeta_{16}^{7} ) q^{52} + ( -4 + 18 \zeta_{16} + 16 \zeta_{16}^{2} - 16 \zeta_{16}^{4} - 18 \zeta_{16}^{5} + 4 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{53} + ( 1 + \zeta_{16} + 8 \zeta_{16}^{2} + 21 \zeta_{16}^{3} - 21 \zeta_{16}^{6} - 8 \zeta_{16}^{7} ) q^{54} + ( 28 - 2 \zeta_{16} + 2 \zeta_{16}^{3} - 28 \zeta_{16}^{4} - 4 \zeta_{16}^{5} + 32 \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{55} + ( -15 + 17 \zeta_{16} + 25 \zeta_{16}^{2} + 25 \zeta_{16}^{3} + 17 \zeta_{16}^{4} - 15 \zeta_{16}^{5} + 33 \zeta_{16}^{6} - 33 \zeta_{16}^{7} ) q^{56} + ( 3 - 6 \zeta_{16} + 10 \zeta_{16}^{2} - 9 \zeta_{16}^{3} + 9 \zeta_{16}^{4} - 10 \zeta_{16}^{5} + 6 \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{57} + ( 30 - 46 \zeta_{16} + 46 \zeta_{16}^{2} - 30 \zeta_{16}^{3} - 4 \zeta_{16}^{4} - 4 \zeta_{16}^{7} ) q^{58} + ( -6 + 6 \zeta_{16}^{2} - 10 \zeta_{16}^{3} + 15 \zeta_{16}^{4} - 42 \zeta_{16}^{5} + 15 \zeta_{16}^{6} - 10 \zeta_{16}^{7} ) q^{59} + ( 8 + 8 \zeta_{16} + 8 \zeta_{16}^{2} - 6 \zeta_{16}^{3} - 6 \zeta_{16}^{4} + 6 \zeta_{16}^{6} + 6 \zeta_{16}^{7} ) q^{60} + ( -20 - 12 \zeta_{16} - 12 \zeta_{16}^{2} - 20 \zeta_{16}^{3} - 8 \zeta_{16}^{4} + 36 \zeta_{16}^{5} - 36 \zeta_{16}^{6} + 8 \zeta_{16}^{7} ) q^{61} + ( -21 + \zeta_{16} - 15 \zeta_{16}^{2} + 7 \zeta_{16}^{3} + 7 \zeta_{16}^{4} - 15 \zeta_{16}^{5} + \zeta_{16}^{6} - 21 \zeta_{16}^{7} ) q^{62} + ( 7 + 21 \zeta_{16} - 15 \zeta_{16}^{2} + 15 \zeta_{16}^{3} - 21 \zeta_{16}^{4} - 7 \zeta_{16}^{5} + 23 \zeta_{16}^{6} + 23 \zeta_{16}^{7} ) q^{63} + ( -8 - 10 \zeta_{16} + 13 \zeta_{16}^{2} - 10 \zeta_{16}^{3} - 8 \zeta_{16}^{4} - 42 \zeta_{16}^{5} + 42 \zeta_{16}^{7} ) q^{64} + ( -12 + 12 \zeta_{16} + 8 \zeta_{16}^{2} + 12 \zeta_{16}^{4} + 12 \zeta_{16}^{5} + 8 \zeta_{16}^{7} ) q^{65} + ( -1 + 23 \zeta_{16} - 12 \zeta_{16}^{2} + 20 \zeta_{16}^{3} - 12 \zeta_{16}^{4} + 23 \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{66} + ( -33 \zeta_{16} + 4 \zeta_{16}^{2} + 26 \zeta_{16}^{3} + 42 \zeta_{16}^{4} + 26 \zeta_{16}^{5} + 4 \zeta_{16}^{6} - 33 \zeta_{16}^{7} ) q^{67} + ( 34 - 34 \zeta_{16}^{2} - 34 \zeta_{16}^{4} + 51 \zeta_{16}^{5} - 34 \zeta_{16}^{6} ) q^{68} + ( 30 + 8 \zeta_{16} + 18 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 18 \zeta_{16}^{6} - 8 \zeta_{16}^{7} ) q^{69} + ( 28 - 16 \zeta_{16} - 60 \zeta_{16}^{2} + 60 \zeta_{16}^{4} + 16 \zeta_{16}^{5} - 28 \zeta_{16}^{6} - 36 \zeta_{16}^{7} ) q^{70} + ( 5 + 5 \zeta_{16} - 43 \zeta_{16}^{2} - 35 \zeta_{16}^{3} - 11 \zeta_{16}^{4} + 11 \zeta_{16}^{5} + 35 \zeta_{16}^{6} + 43 \zeta_{16}^{7} ) q^{71} + ( 5 - \zeta_{16} + \zeta_{16}^{3} - 5 \zeta_{16}^{4} + 24 \zeta_{16}^{5} - 53 \zeta_{16}^{6} + 24 \zeta_{16}^{7} ) q^{72} + ( 6 - 6 \zeta_{16} + 25 \zeta_{16}^{2} + 25 \zeta_{16}^{3} - 6 \zeta_{16}^{4} + 6 \zeta_{16}^{5} - 26 \zeta_{16}^{6} + 26 \zeta_{16}^{7} ) q^{73} + ( -20 + 26 \zeta_{16} + 28 \zeta_{16}^{2} - 24 \zeta_{16}^{3} + 24 \zeta_{16}^{4} - 28 \zeta_{16}^{5} - 26 \zeta_{16}^{6} + 20 \zeta_{16}^{7} ) q^{74} + ( -37 + 17 \zeta_{16} - 17 \zeta_{16}^{2} + 37 \zeta_{16}^{3} + 13 \zeta_{16}^{5} + 13 \zeta_{16}^{6} ) q^{75} + ( 10 - 10 \zeta_{16}^{2} + 21 \zeta_{16}^{3} - 20 \zeta_{16}^{4} + 12 \zeta_{16}^{5} - 20 \zeta_{16}^{6} + 21 \zeta_{16}^{7} ) q^{76} + ( -6 - 66 \zeta_{16} - 6 \zeta_{16}^{2} + 8 \zeta_{16}^{3} - 10 \zeta_{16}^{4} + 10 \zeta_{16}^{6} - 8 \zeta_{16}^{7} ) q^{77} + ( -50 - 8 \zeta_{16} - 8 \zeta_{16}^{2} - 50 \zeta_{16}^{3} + 38 \zeta_{16}^{4} - 64 \zeta_{16}^{5} + 64 \zeta_{16}^{6} - 38 \zeta_{16}^{7} ) q^{78} + ( -17 + 41 \zeta_{16} - 3 \zeta_{16}^{2} - 21 \zeta_{16}^{3} - 21 \zeta_{16}^{4} - 3 \zeta_{16}^{5} + 41 \zeta_{16}^{6} - 17 \zeta_{16}^{7} ) q^{79} + ( -30 - 14 \zeta_{16} + 8 \zeta_{16}^{2} - 8 \zeta_{16}^{3} + 14 \zeta_{16}^{4} + 30 \zeta_{16}^{5} - 30 \zeta_{16}^{6} - 30 \zeta_{16}^{7} ) q^{80} + ( -53 + 6 \zeta_{16} - 3 \zeta_{16}^{2} + 6 \zeta_{16}^{3} - 53 \zeta_{16}^{4} + 7 \zeta_{16}^{5} - 7 \zeta_{16}^{7} ) q^{81} + ( -8 + 8 \zeta_{16} + 8 \zeta_{16}^{2} + 34 \zeta_{16}^{3} - 17 \zeta_{16}^{4} - 17 \zeta_{16}^{5} + 34 \zeta_{16}^{6} + 8 \zeta_{16}^{7} ) q^{82} + ( -33 + 20 \zeta_{16} + 42 \zeta_{16}^{2} + 54 \zeta_{16}^{3} + 42 \zeta_{16}^{4} + 20 \zeta_{16}^{5} - 33 \zeta_{16}^{6} ) q^{83} + ( -34 \zeta_{16} + 24 \zeta_{16}^{2} - 60 \zeta_{16}^{3} + 50 \zeta_{16}^{4} - 60 \zeta_{16}^{5} + 24 \zeta_{16}^{6} - 34 \zeta_{16}^{7} ) q^{84} + ( -34 + 34 \zeta_{16} + 34 \zeta_{16}^{2} - 34 \zeta_{16}^{3} - 34 \zeta_{16}^{4} + 34 \zeta_{16}^{7} ) q^{85} + ( 104 - 66 \zeta_{16} - 8 \zeta_{16}^{2} - 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} + 8 \zeta_{16}^{6} + 66 \zeta_{16}^{7} ) q^{86} + ( 26 - 34 \zeta_{16} + 10 \zeta_{16}^{2} - 10 \zeta_{16}^{4} + 34 \zeta_{16}^{5} - 26 \zeta_{16}^{6} + 52 \zeta_{16}^{7} ) q^{87} + ( 33 + 33 \zeta_{16} - \zeta_{16}^{2} - 9 \zeta_{16}^{3} + 40 \zeta_{16}^{4} - 40 \zeta_{16}^{5} + 9 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{88} + ( 20 - 46 \zeta_{16} + 46 \zeta_{16}^{3} - 20 \zeta_{16}^{4} - 29 \zeta_{16}^{5} - 8 \zeta_{16}^{6} - 29 \zeta_{16}^{7} ) q^{89} + ( 56 - 18 \zeta_{16} - 28 \zeta_{16}^{2} - 28 \zeta_{16}^{3} - 18 \zeta_{16}^{4} + 56 \zeta_{16}^{5} + 48 \zeta_{16}^{6} - 48 \zeta_{16}^{7} ) q^{90} + ( 40 + 68 \zeta_{16} + 28 \zeta_{16}^{3} - 28 \zeta_{16}^{4} - 68 \zeta_{16}^{6} - 40 \zeta_{16}^{7} ) q^{91} + ( 3 + 31 \zeta_{16} - 31 \zeta_{16}^{2} - 3 \zeta_{16}^{3} - 23 \zeta_{16}^{4} - 3 \zeta_{16}^{5} - 3 \zeta_{16}^{6} - 23 \zeta_{16}^{7} ) q^{92} + ( -8 + 8 \zeta_{16}^{2} - 8 \zeta_{16}^{3} + 26 \zeta_{16}^{4} - 6 \zeta_{16}^{5} + 26 \zeta_{16}^{6} - 8 \zeta_{16}^{7} ) q^{93} + ( 4 + 110 \zeta_{16} + 4 \zeta_{16}^{2} - 18 \zeta_{16}^{3} - 54 \zeta_{16}^{4} + 54 \zeta_{16}^{6} + 18 \zeta_{16}^{7} ) q^{94} + ( 34 - 24 \zeta_{16} - 24 \zeta_{16}^{2} + 34 \zeta_{16}^{3} - 2 \zeta_{16}^{4} - 24 \zeta_{16}^{5} + 24 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{95} + ( -7 + 8 \zeta_{16} + 11 \zeta_{16}^{2} + 29 \zeta_{16}^{3} + 29 \zeta_{16}^{4} + 11 \zeta_{16}^{5} + 8 \zeta_{16}^{6} - 7 \zeta_{16}^{7} ) q^{96} + ( 6 + 13 \zeta_{16} + 30 \zeta_{16}^{2} - 30 \zeta_{16}^{3} - 13 \zeta_{16}^{4} - 6 \zeta_{16}^{5} - 22 \zeta_{16}^{6} - 22 \zeta_{16}^{7} ) q^{97} + ( -15 - 11 \zeta_{16} + 9 \zeta_{16}^{2} - 11 \zeta_{16}^{3} - 15 \zeta_{16}^{4} - 9 \zeta_{16}^{5} + 9 \zeta_{16}^{7} ) q^{98} + ( -28 + 28 \zeta_{16} - 16 \zeta_{16}^{2} - 27 \zeta_{16}^{3} + 34 \zeta_{16}^{4} + 34 \zeta_{16}^{5} - 27 \zeta_{16}^{6} - 16 \zeta_{16}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{2} - 8q^{3} + 16q^{5} - 8q^{6} + 8q^{7} - 24q^{8} - 16q^{9} + O(q^{10}) \) \( 8q - 8q^{2} - 8q^{3} + 16q^{5} - 8q^{6} + 8q^{7} - 24q^{8} - 16q^{9} + 16q^{10} - 8q^{11} + 48q^{12} + 16q^{13} + 8q^{14} - 16q^{15} + 56q^{18} - 80q^{20} - 64q^{21} - 104q^{22} - 56q^{23} - 80q^{24} + 64q^{25} + 176q^{26} + 40q^{27} + 152q^{28} + 48q^{29} + 16q^{30} + 24q^{31} + 88q^{32} - 136q^{34} - 160q^{35} - 128q^{36} + 32q^{37} - 120q^{38} + 48q^{39} + 64q^{40} + 48q^{41} + 16q^{42} - 232q^{43} + 120q^{44} - 88q^{46} + 192q^{47} + 136q^{48} + 16q^{49} + 136q^{51} - 384q^{52} - 32q^{53} + 8q^{54} + 224q^{55} - 120q^{56} + 24q^{57} + 240q^{58} - 48q^{59} + 64q^{60} - 160q^{61} - 168q^{62} + 56q^{63} - 64q^{64} - 96q^{65} - 8q^{66} + 272q^{68} + 240q^{69} + 224q^{70} + 40q^{71} + 40q^{72} + 48q^{73} - 160q^{74} - 296q^{75} + 80q^{76} - 48q^{77} - 400q^{78} - 136q^{79} - 240q^{80} - 424q^{81} - 64q^{82} - 264q^{83} - 272q^{85} + 832q^{86} + 208q^{87} + 264q^{88} + 160q^{89} + 448q^{90} + 320q^{91} + 24q^{92} - 64q^{93} + 32q^{94} + 272q^{95} - 56q^{96} + 48q^{97} - 120q^{98} - 224q^{99} + O(q^{100}) \)

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
1.23044 0.509666i −2.63099 0.523336i −1.57420 + 1.57420i 0.902812 + 1.35115i −3.50400 + 0.696990i 4.92562 7.37170i −3.17331 + 7.66104i −1.66671 0.690373i 1.79949 + 1.20238i
5.1 −0.841487 2.03153i 0.0897902 + 0.134381i −0.590587 + 0.590587i −1.04667 + 5.26197i 0.197441 0.295491i 1.21824 + 6.12453i −6.42935 2.66313i 3.43416 8.29078i 11.5706 2.30154i
6.1 1.23044 + 0.509666i −2.63099 + 0.523336i −1.57420 1.57420i 0.902812 1.35115i −3.50400 0.696990i 4.92562 + 7.37170i −3.17331 7.66104i −1.66671 + 0.690373i 1.79949 1.20238i
7.1 −0.841487 + 2.03153i 0.0897902 0.134381i −0.590587 0.590587i −1.04667 5.26197i 0.197441 + 0.295491i 1.21824 6.12453i −6.42935 + 2.66313i 3.43416 + 8.29078i 11.5706 + 2.30154i
10.1 −1.15851 + 2.79690i −0.675577 0.451406i −3.65205 3.65205i 7.87510 1.56645i 2.04520 1.36656i −7.70353 1.53233i 3.25778 1.34942i −3.19151 7.70500i −4.74219 + 23.8406i
11.1 −3.23044 1.33809i −0.783227 3.93755i 5.81684 + 5.81684i 0.268761 + 0.179580i −2.73864 + 13.7681i 5.55967 3.71485i −5.65512 13.6527i −6.57593 + 2.72384i −0.627922 0.939752i
12.1 −1.15851 2.79690i −0.675577 + 0.451406i −3.65205 + 3.65205i 7.87510 + 1.56645i 2.04520 + 1.36656i −7.70353 + 1.53233i 3.25778 + 1.34942i −3.19151 + 7.70500i −4.74219 23.8406i
14.1 −3.23044 + 1.33809i −0.783227 + 3.93755i 5.81684 5.81684i 0.268761 0.179580i −2.73864 13.7681i 5.55967 + 3.71485i −5.65512 + 13.6527i −6.57593 2.72384i −0.627922 + 0.939752i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.1
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{3}^{\mathrm{new}}(17, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 8 T + 32 T^{2} + 104 T^{3} + 320 T^{4} + 872 T^{5} + 2112 T^{6} + 4776 T^{7} + 10017 T^{8} + 19104 T^{9} + 33792 T^{10} + 55808 T^{11} + 81920 T^{12} + 106496 T^{13} + 131072 T^{14} + 131072 T^{15} + 65536 T^{16} \)
$3$ \( 1 + 8 T + 40 T^{2} + 136 T^{3} + 530 T^{4} + 1976 T^{5} + 7064 T^{6} + 19576 T^{7} + 58466 T^{8} + 176184 T^{9} + 572184 T^{10} + 1440504 T^{11} + 3477330 T^{12} + 8030664 T^{13} + 21257640 T^{14} + 38263752 T^{15} + 43046721 T^{16} \)
$5$ \( 1 - 16 T + 96 T^{2} - 112 T^{3} - 352 T^{4} - 10432 T^{5} + 95824 T^{6} - 124272 T^{7} - 914688 T^{8} - 3106800 T^{9} + 59890000 T^{10} - 163000000 T^{11} - 137500000 T^{12} - 1093750000 T^{13} + 23437500000 T^{14} - 97656250000 T^{15} + 152587890625 T^{16} \)
$7$ \( 1 - 8 T + 24 T^{2} - 1128 T^{3} + 8240 T^{4} - 20216 T^{5} + 640360 T^{6} - 4599368 T^{7} + 10019136 T^{8} - 225369032 T^{9} + 1537504360 T^{10} - 2378392184 T^{11} + 47501960240 T^{12} - 318632080872 T^{13} + 332190892824 T^{14} - 5425784582792 T^{15} + 33232930569601 T^{16} \)
$11$ \( 1 + 8 T + 12 T^{2} + 536 T^{3} - 10730 T^{4} - 47224 T^{5} + 541212 T^{6} - 224040 T^{7} + 130587298 T^{8} - 27108840 T^{9} + 7923884892 T^{10} - 83660196664 T^{11} - 2300070793130 T^{12} + 13902459586136 T^{13} + 37661140520652 T^{14} + 3037998668665928 T^{15} + 45949729863572161 T^{16} \)
$13$ \( 1 - 16 T + 128 T^{2} - 16 T^{3} - 5116 T^{4} + 117072 T^{5} - 1218176 T^{6} + 73648208 T^{7} - 1288359866 T^{8} + 12446547152 T^{9} - 34792324736 T^{10} + 565084183248 T^{11} - 4173278368636 T^{12} - 2205735869584 T^{13} + 2982154895677568 T^{14} - 62998022171188624 T^{15} + 665416609183179841 T^{16} \)
$17$ \( 1 + 6975757441 T^{8} \)
$19$ \( 1 + 368 T^{2} + 7440 T^{3} + 67712 T^{4} + 1902864 T^{5} + 27896560 T^{6} - 410926464 T^{7} + 18992960962 T^{8} - 148344453504 T^{9} + 3635507595760 T^{10} + 89521913303184 T^{11} + 1149991020632192 T^{12} + 45615132958039440 T^{13} + 814499890216347248 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \)
$23$ \( 1 + 56 T + 1032 T^{2} - 18712 T^{3} - 1041904 T^{4} - 8737672 T^{5} + 292851208 T^{6} + 8265977256 T^{7} + 119476335168 T^{8} + 4372701968424 T^{9} + 81951774897928 T^{10} - 1293489042310408 T^{11} - 81592528808215024 T^{12} - 775172877829800088 T^{13} + 22615892413844971272 T^{14} + \)\(64\!\cdots\!04\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} \)
$29$ \( 1 - 48 T + 624 T^{2} + 56960 T^{3} - 3050368 T^{4} + 68290448 T^{5} + 241251280 T^{6} - 64889435200 T^{7} + 2514250952448 T^{8} - 54572015003200 T^{9} + 170632446569680 T^{10} + 40620751071937808 T^{11} - 1525935650211019648 T^{12} + 23963484008779448960 T^{13} + \)\(22\!\cdots\!84\)\( T^{14} - \)\(14\!\cdots\!88\)\( T^{15} + \)\(25\!\cdots\!21\)\( T^{16} \)
$31$ \( 1 - 24 T + 232 T^{2} + 33928 T^{3} - 415280 T^{4} - 30121928 T^{5} + 884607752 T^{6} - 20180029736 T^{7} - 595898026688 T^{8} - 19393008576296 T^{9} + 816953835734792 T^{10} - 26733321978816968 T^{11} - 354188590028498480 T^{12} + 27808348520684616328 T^{13} + \)\(18\!\cdots\!52\)\( T^{14} - \)\(18\!\cdots\!04\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} \)
$37$ \( 1 - 32 T + 1024 T^{2} + 43232 T^{3} - 3988832 T^{4} + 114777552 T^{5} - 1479427536 T^{6} - 166606879104 T^{7} + 7610311129088 T^{8} - 228084817493376 T^{9} - 2772685390297296 T^{10} + 294487796326770768 T^{11} - 14010690445142610272 T^{12} + \)\(20\!\cdots\!68\)\( T^{13} + \)\(67\!\cdots\!44\)\( T^{14} - \)\(28\!\cdots\!48\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \)
$41$ \( 1 - 48 T + 1824 T^{2} + 23984 T^{3} - 1206974 T^{4} + 18567760 T^{5} + 1403379552 T^{6} - 142523024592 T^{7} + 3825436575874 T^{8} - 239581204339152 T^{9} + 3965615206239072 T^{10} + 88198795521870160 T^{11} - 9637597143493089854 T^{12} + \)\(32\!\cdots\!84\)\( T^{13} + \)\(41\!\cdots\!44\)\( T^{14} - \)\(18\!\cdots\!28\)\( T^{15} + \)\(63\!\cdots\!41\)\( T^{16} \)
$43$ \( 1 + 232 T + 27932 T^{2} + 2342968 T^{3} + 154510664 T^{4} + 8541190600 T^{5} + 413867884116 T^{6} + 18437331684504 T^{7} + 795437538561102 T^{8} + 34090626284647896 T^{9} + 1414931936083664916 T^{10} + 53991966653306139400 T^{11} + \)\(18\!\cdots\!64\)\( T^{12} + \)\(50\!\cdots\!32\)\( T^{13} + \)\(11\!\cdots\!32\)\( T^{14} + \)\(17\!\cdots\!68\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - 192 T + 18432 T^{2} - 1331200 T^{3} + 86363396 T^{4} - 5070960256 T^{5} + 267820974080 T^{6} - 13081254629824 T^{7} + 617534815843590 T^{8} - 28896491477281216 T^{9} + 1306880918619668480 T^{10} - 54660972524224964224 T^{11} + \)\(20\!\cdots\!56\)\( T^{12} - \)\(70\!\cdots\!00\)\( T^{13} + \)\(21\!\cdots\!12\)\( T^{14} - \)\(49\!\cdots\!48\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( 1 + 32 T + 2272 T^{2} + 283520 T^{3} + 9589248 T^{4} + 703635840 T^{5} + 40677888352 T^{6} + 1429537588256 T^{7} + 116127856966210 T^{8} + 4015571085411104 T^{9} + 320968105161577312 T^{10} + 15595638861067263360 T^{11} + \)\(59\!\cdots\!28\)\( T^{12} + \)\(49\!\cdots\!80\)\( T^{13} + \)\(11\!\cdots\!52\)\( T^{14} + \)\(44\!\cdots\!08\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \)
$59$ \( 1 + 48 T + 5988 T^{2} + 249936 T^{3} + 17455752 T^{4} - 267650832 T^{5} - 2258950740 T^{6} - 4680295454832 T^{7} - 153985341695026 T^{8} - 16292108478270192 T^{9} - 27372521597797140 T^{10} - 11289654923217639312 T^{11} + \)\(25\!\cdots\!92\)\( T^{12} + \)\(12\!\cdots\!36\)\( T^{13} + \)\(10\!\cdots\!28\)\( T^{14} + \)\(29\!\cdots\!28\)\( T^{15} + \)\(21\!\cdots\!41\)\( T^{16} \)
$61$ \( 1 + 160 T + 8384 T^{2} - 591392 T^{3} - 112289792 T^{4} - 7004799072 T^{5} - 46635351424 T^{6} + 25101045742144 T^{7} + 2320428818600960 T^{8} + 93400991206517824 T^{9} - 645705660795827584 T^{10} - \)\(36\!\cdots\!92\)\( T^{11} - \)\(21\!\cdots\!52\)\( T^{12} - \)\(42\!\cdots\!92\)\( T^{13} + \)\(22\!\cdots\!64\)\( T^{14} + \)\(15\!\cdots\!60\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} \)
$67$ \( 1 - 14608 T^{2} + 108494176 T^{4} - 545817039504 T^{6} + 2442127301002178 T^{8} - 10998825206906883984 T^{10} + \)\(44\!\cdots\!16\)\( T^{12} - \)\(11\!\cdots\!88\)\( T^{14} + \)\(16\!\cdots\!81\)\( T^{16} \)
$71$ \( 1 - 40 T - 12216 T^{2} - 301752 T^{3} + 88296464 T^{4} + 4708166648 T^{5} - 160406696312 T^{6} - 17457547717336 T^{7} - 106016902560192 T^{8} - 88003498043090776 T^{9} - 4076203796944420472 T^{10} + \)\(60\!\cdots\!08\)\( T^{11} + \)\(57\!\cdots\!04\)\( T^{12} - \)\(98\!\cdots\!52\)\( T^{13} - \)\(20\!\cdots\!56\)\( T^{14} - \)\(33\!\cdots\!40\)\( T^{15} + \)\(41\!\cdots\!21\)\( T^{16} \)
$73$ \( 1 - 48 T - 4096 T^{2} - 616528 T^{3} + 30492930 T^{4} + 4872133264 T^{5} + 204783337600 T^{6} - 9842384223888 T^{7} - 2875677804870654 T^{8} - 52450065529099152 T^{9} + 5815486573949161600 T^{10} + \)\(73\!\cdots\!96\)\( T^{11} + \)\(24\!\cdots\!30\)\( T^{12} - \)\(26\!\cdots\!72\)\( T^{13} - \)\(93\!\cdots\!16\)\( T^{14} - \)\(58\!\cdots\!32\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \)
$79$ \( 1 + 136 T + 3800 T^{2} - 1305896 T^{3} - 161585328 T^{4} - 6145771400 T^{5} + 317342984696 T^{6} + 54517171591464 T^{7} + 4138695597908032 T^{8} + 340241667902326824 T^{9} + 12360534958690960376 T^{10} - \)\(14\!\cdots\!00\)\( T^{11} - \)\(24\!\cdots\!08\)\( T^{12} - \)\(12\!\cdots\!96\)\( T^{13} + \)\(22\!\cdots\!00\)\( T^{14} + \)\(50\!\cdots\!16\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} \)
$83$ \( 1 + 264 T + 35100 T^{2} + 2811288 T^{3} + 126238536 T^{4} + 5231440488 T^{5} + 1212641715924 T^{6} + 223465365584760 T^{7} + 23797706896032974 T^{8} + 1539452903513411640 T^{9} + 57549939812312003604 T^{10} + \)\(17\!\cdots\!72\)\( T^{11} + \)\(28\!\cdots\!76\)\( T^{12} + \)\(43\!\cdots\!12\)\( T^{13} + \)\(37\!\cdots\!00\)\( T^{14} + \)\(19\!\cdots\!56\)\( T^{15} + \)\(50\!\cdots\!81\)\( T^{16} \)
$89$ \( 1 - 160 T + 12800 T^{2} - 1203232 T^{3} + 97134272 T^{4} - 4832684576 T^{5} + 253794473472 T^{6} + 8995658901856 T^{7} - 3993981113240190 T^{8} + 71254614161601376 T^{9} + 15923634019048330752 T^{10} - \)\(24\!\cdots\!36\)\( T^{11} + \)\(38\!\cdots\!32\)\( T^{12} - \)\(37\!\cdots\!32\)\( T^{13} + \)\(31\!\cdots\!00\)\( T^{14} - \)\(31\!\cdots\!60\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} \)
$97$ \( 1 - 48 T - 4444 T^{2} - 1737360 T^{3} + 80511142 T^{4} + 14428723088 T^{5} + 1714599118564 T^{6} - 120003485930960 T^{7} - 19840415339669054 T^{8} - 1129112799124402640 T^{9} + \)\(15\!\cdots\!84\)\( T^{10} + \)\(12\!\cdots\!52\)\( T^{11} + \)\(63\!\cdots\!62\)\( T^{12} - \)\(12\!\cdots\!40\)\( T^{13} - \)\(30\!\cdots\!04\)\( T^{14} - \)\(31\!\cdots\!12\)\( T^{15} + \)\(61\!\cdots\!21\)\( T^{16} \)
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