# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{3}^{\mathrm{new}}(17, [\chi])$$.