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 \(16\) and degree \(8\))

Newform invariants

Self dual: No
Analytic conductor: \(0.463216449413\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
17.e Odd 1 yes

Hecke kernels

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