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

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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.

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

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
−1.63099 + 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:

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 \(T_{2}^{8} \) \(\mathstrut +\mathstrut 4 T_{2}^{6} \) \(\mathstrut +\mathstrut 16 T_{2}^{5} \) \(\mathstrut +\mathstrut 8 T_{2}^{4} \) \(\mathstrut -\mathstrut 8 T_{2}^{3} \) \(\mathstrut +\mathstrut 20 T_{2}^{2} \) \(\mathstrut -\mathstrut 8 T_{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{3}^{\mathrm{new}}(17, [\chi])\).