# Properties

 Label 169.4.a.k Level $169$ Weight $4$ Character orbit 169.a Self dual yes Analytic conductor $9.971$ Analytic rank $1$ Dimension $9$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.97132279097$$ Analytic rank: $$1$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - 4 x^{8} - 46 x^{7} + 145 x^{6} + 680 x^{5} - 1501 x^{4} - 3203 x^{3} + 4784 x^{2} + 3584 x - 4096$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$13^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + \beta_{2} q^{3} + ( 5 + \beta_{5} + \beta_{6} + \beta_{8} ) q^{4} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} ) q^{5} + ( -6 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{6} - 2 \beta_{8} ) q^{6} + ( -4 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{8} ) q^{7} + ( -9 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{5} ) q^{8} + ( 7 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + \beta_{2} q^{3} + ( 5 + \beta_{5} + \beta_{6} + \beta_{8} ) q^{4} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} ) q^{5} + ( -6 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{6} - 2 \beta_{8} ) q^{6} + ( -4 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{8} ) q^{7} + ( -9 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{5} ) q^{8} + ( 7 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{9} + ( -16 - 6 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{10} + ( -21 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{11} + ( 10 - 14 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{12} + ( -10 - 9 \beta_{1} - \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} ) q^{14} + ( -20 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + \beta_{5} + 9 \beta_{6} + \beta_{7} + \beta_{8} ) q^{15} + ( 37 - 7 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} - 5 \beta_{4} + \beta_{5} + 7 \beta_{6} + 2 \beta_{8} ) q^{16} + ( -7 - 8 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} + \beta_{5} - 7 \beta_{6} + 5 \beta_{7} - 7 \beta_{8} ) q^{17} + ( -5 + 3 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - 5 \beta_{4} + \beta_{5} + 9 \beta_{6} - 2 \beta_{7} + 5 \beta_{8} ) q^{18} + ( -14 - 4 \beta_{1} + \beta_{2} + 3 \beta_{5} - \beta_{6} + \beta_{7} + 10 \beta_{8} ) q^{19} + ( -36 - 15 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} + 10 \beta_{4} - 3 \beta_{5} - 5 \beta_{6} - 6 \beta_{7} - 9 \beta_{8} ) q^{20} + ( -24 + 4 \beta_{1} + 2 \beta_{2} + 14 \beta_{3} - 5 \beta_{4} + 10 \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{21} + ( 43 - 15 \beta_{1} + 10 \beta_{2} + 3 \beta_{3} + 8 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} + 3 \beta_{8} ) q^{22} + ( -22 + 8 \beta_{1} - 6 \beta_{2} + \beta_{4} + 14 \beta_{5} + 2 \beta_{6} - 3 \beta_{8} ) q^{23} + ( -94 - 3 \beta_{1} + 11 \beta_{2} + 5 \beta_{3} + \beta_{4} - 9 \beta_{5} - 7 \beta_{6} + 6 \beta_{7} + 6 \beta_{8} ) q^{24} + ( 25 + 23 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + \beta_{6} - 7 \beta_{7} + 3 \beta_{8} ) q^{25} + ( -83 + 29 \beta_{1} + 6 \beta_{2} - \beta_{3} - 4 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 19 \beta_{8} ) q^{27} + ( -46 + \beta_{1} + 4 \beta_{2} + 10 \beta_{3} - 25 \beta_{4} + 12 \beta_{5} + 8 \beta_{7} - 5 \beta_{8} ) q^{28} + ( 16 + 29 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - 5 \beta_{5} + 6 \beta_{6} - 8 \beta_{8} ) q^{29} + ( 50 + 22 \beta_{1} - 9 \beta_{2} + 34 \beta_{3} - 16 \beta_{4} + 6 \beta_{5} + 11 \beta_{6} + 8 \beta_{7} + 17 \beta_{8} ) q^{30} + ( -82 + 21 \beta_{1} + 7 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 12 \beta_{6} - 8 \beta_{7} - 4 \beta_{8} ) q^{31} + ( -89 + 18 \beta_{1} + 17 \beta_{2} + 10 \beta_{3} + 7 \beta_{4} - 18 \beta_{5} + \beta_{6} - 10 \beta_{7} - 12 \beta_{8} ) q^{32} + ( -59 + 24 \beta_{1} - 23 \beta_{2} - 22 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} ) q^{33} + ( -19 - 37 \beta_{1} + 10 \beta_{2} - 23 \beta_{3} + 25 \beta_{4} - 6 \beta_{5} - 29 \beta_{6} + 10 \beta_{7} - 9 \beta_{8} ) q^{34} + ( -20 + 40 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} - \beta_{4} - 4 \beta_{5} - 17 \beta_{6} + \beta_{7} + 8 \beta_{8} ) q^{35} + ( -115 + 31 \beta_{1} + 12 \beta_{2} + 16 \beta_{3} + 2 \beta_{4} - 12 \beta_{5} + 12 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} ) q^{36} + ( -12 - 28 \beta_{1} - 16 \beta_{2} + 14 \beta_{3} + 9 \beta_{4} + 4 \beta_{5} + 13 \beta_{6} - 11 \beta_{7} - 12 \beta_{8} ) q^{37} + ( -90 - 4 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} + 20 \beta_{4} - 8 \beta_{5} - 21 \beta_{6} - 18 \beta_{8} ) q^{38} + ( -8 - 18 \beta_{1} - 9 \beta_{2} - 24 \beta_{3} - 22 \beta_{4} + 16 \beta_{5} - 35 \beta_{6} + 4 \beta_{7} - 15 \beta_{8} ) q^{40} + ( -150 + 14 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 21 \beta_{5} - 12 \beta_{6} + 2 \beta_{7} + 7 \beta_{8} ) q^{41} + ( 20 - 6 \beta_{1} - 2 \beta_{2} - 26 \beta_{3} + 25 \beta_{4} - 5 \beta_{5} + 19 \beta_{6} - 10 \beta_{7} - 2 \beta_{8} ) q^{42} + ( -70 + 14 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 25 \beta_{4} + 21 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} - 11 \beta_{8} ) q^{43} + ( -63 - 7 \beta_{2} - 15 \beta_{3} - 11 \beta_{4} - 17 \beta_{5} - 29 \beta_{6} + 16 \beta_{7} - 15 \beta_{8} ) q^{44} + ( 26 - 42 \beta_{1} - 42 \beta_{2} - 14 \beta_{3} + 13 \beta_{4} + 8 \beta_{5} - 20 \beta_{6} + 10 \beta_{7} - 11 \beta_{8} ) q^{45} + ( 82 + 39 \beta_{1} + 20 \beta_{2} + 26 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} + 12 \beta_{6} + 2 \beta_{7} + 27 \beta_{8} ) q^{46} + ( -92 - 25 \beta_{1} - 23 \beta_{2} + \beta_{3} - 25 \beta_{4} + 7 \beta_{5} + 10 \beta_{6} + 4 \beta_{7} + 5 \beta_{8} ) q^{47} + ( 6 - 42 \beta_{1} - 9 \beta_{2} - 33 \beta_{3} + 10 \beta_{4} + 23 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 37 \beta_{8} ) q^{48} + ( 55 - 25 \beta_{1} - 5 \beta_{2} - 41 \beta_{3} + 39 \beta_{4} - 27 \beta_{5} - 19 \beta_{6} + 7 \beta_{7} + 6 \beta_{8} ) q^{49} + ( 113 + 53 \beta_{1} - 9 \beta_{2} - 2 \beta_{3} - 26 \beta_{4} + 22 \beta_{5} + 11 \beta_{6} - 8 \beta_{7} + 5 \beta_{8} ) q^{50} + ( 49 + 44 \beta_{1} - 11 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 24 \beta_{5} + 40 \beta_{6} + 8 \beta_{7} + 30 \beta_{8} ) q^{51} + ( -18 - 18 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} + 36 \beta_{4} - 6 \beta_{5} + 13 \beta_{6} + 3 \beta_{7} - 43 \beta_{8} ) q^{53} + ( 325 - 49 \beta_{1} - 32 \beta_{2} + 8 \beta_{3} + 10 \beta_{4} + 26 \beta_{5} + 18 \beta_{6} - 8 \beta_{7} - 5 \beta_{8} ) q^{54} + ( -50 + 37 \beta_{1} + 19 \beta_{2} + 83 \beta_{3} - 38 \beta_{4} + 19 \beta_{5} + 7 \beta_{6} + 7 \beta_{7} + 37 \beta_{8} ) q^{55} + ( 30 - 4 \beta_{1} + 36 \beta_{2} - 4 \beta_{3} + 55 \beta_{4} - 35 \beta_{5} - \beta_{6} - 18 \beta_{7} - 18 \beta_{8} ) q^{56} + ( 113 - 34 \beta_{1} - 43 \beta_{2} + 40 \beta_{3} - 4 \beta_{4} + 21 \beta_{5} - \beta_{6} - \beta_{7} + 18 \beta_{8} ) q^{57} + ( 346 + 25 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - 16 \beta_{4} + 25 \beta_{5} + 60 \beta_{6} - 12 \beta_{7} + 28 \beta_{8} ) q^{58} + ( -158 - 49 \beta_{1} + 30 \beta_{2} + \beta_{3} + 8 \beta_{4} - 13 \beta_{5} - 22 \beta_{6} + 22 \beta_{7} - 34 \beta_{8} ) q^{59} + ( 354 + 85 \beta_{1} - 23 \beta_{2} - 4 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} + 51 \beta_{6} - 40 \beta_{7} + 20 \beta_{8} ) q^{60} + ( 4 + 6 \beta_{1} + 16 \beta_{2} + 52 \beta_{3} + 17 \beta_{4} - 18 \beta_{5} + 51 \beta_{6} - 13 \beta_{7} + 18 \beta_{8} ) q^{61} + ( 248 - 27 \beta_{1} - 17 \beta_{2} + 38 \beta_{3} - 72 \beta_{4} + 33 \beta_{5} + 32 \beta_{6} + 4 \beta_{7} + 34 \beta_{8} ) q^{62} + ( 86 + 27 \beta_{1} + 27 \beta_{2} - 15 \beta_{3} + 34 \beta_{4} - 33 \beta_{5} - 15 \beta_{6} + 13 \beta_{7} + 17 \beta_{8} ) q^{63} + ( 49 - 71 \beta_{1} - 9 \beta_{2} + 22 \beta_{3} - 66 \beta_{4} + 62 \beta_{5} - 27 \beta_{6} + 14 \beta_{7} + 15 \beta_{8} ) q^{64} + ( 381 - 7 \beta_{1} + 39 \beta_{2} + 79 \beta_{3} + 2 \beta_{4} + \beta_{5} + 33 \beta_{6} - 8 \beta_{7} + 43 \beta_{8} ) q^{66} + ( 102 + 45 \beta_{1} + 28 \beta_{2} - 57 \beta_{3} + 37 \beta_{4} - 3 \beta_{5} - 9 \beta_{6} - 31 \beta_{7} + 22 \beta_{8} ) q^{67} + ( -185 - 76 \beta_{1} + 4 \beta_{2} - 51 \beta_{3} + 26 \beta_{4} - 22 \beta_{5} - 115 \beta_{6} + 10 \beta_{7} - 4 \beta_{8} ) q^{68} + ( -212 - 15 \beta_{1} - 39 \beta_{2} - 101 \beta_{3} + 15 \beta_{4} - 57 \beta_{5} - 10 \beta_{6} - 6 \beta_{7} + 27 \beta_{8} ) q^{69} + ( 576 - 26 \beta_{1} + 30 \beta_{2} - 20 \beta_{3} + 57 \beta_{4} + 29 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + 24 \beta_{8} ) q^{70} + ( -264 + 11 \beta_{1} + 71 \beta_{2} - 11 \beta_{3} - 27 \beta_{4} + 17 \beta_{5} - 27 \beta_{6} + 23 \beta_{7} - 26 \beta_{8} ) q^{71} + ( 543 - 117 \beta_{1} - 67 \beta_{2} + 30 \beta_{3} - 29 \beta_{4} + 57 \beta_{5} + 4 \beta_{6} + 20 \beta_{7} + 4 \beta_{8} ) q^{72} + ( -52 + 85 \beta_{1} + 50 \beta_{2} + 5 \beta_{3} - 45 \beta_{4} - 39 \beta_{5} + 19 \beta_{6} + 15 \beta_{7} + 2 \beta_{8} ) q^{73} + ( -232 + 44 \beta_{1} - 4 \beta_{2} + 42 \beta_{3} - 121 \beta_{4} + 11 \beta_{5} + 57 \beta_{6} + 18 \beta_{7} + 72 \beta_{8} ) q^{74} + ( 226 + 15 \beta_{1} + 36 \beta_{2} - 3 \beta_{3} - 18 \beta_{4} + 11 \beta_{5} - 64 \beta_{6} - 36 \beta_{7} - 34 \beta_{8} ) q^{75} + ( 434 - 120 \beta_{1} + 12 \beta_{2} - 60 \beta_{3} - 22 \beta_{4} + 18 \beta_{5} - 22 \beta_{6} + 32 \beta_{7} - 35 \beta_{8} ) q^{76} + ( -32 - 40 \beta_{1} + 24 \beta_{2} - 50 \beta_{3} + 31 \beta_{4} + 22 \beta_{5} + 65 \beta_{6} - 15 \beta_{7} - 76 \beta_{8} ) q^{77} + ( 44 - 5 \beta_{1} - 3 \beta_{2} + 77 \beta_{3} + 3 \beta_{4} + 9 \beta_{5} - 5 \beta_{6} + 21 \beta_{7} + 38 \beta_{8} ) q^{79} + ( 56 - 17 \beta_{1} + 57 \beta_{2} + 12 \beta_{3} + 79 \beta_{4} - 82 \beta_{5} - 57 \beta_{6} + 4 \beta_{7} - 20 \beta_{8} ) q^{80} + ( -62 - 124 \beta_{1} - 85 \beta_{2} - 8 \beta_{3} + 27 \beta_{4} + 74 \beta_{5} - 72 \beta_{6} + 32 \beta_{7} - 81 \beta_{8} ) q^{81} + ( 438 - 237 \beta_{1} - 23 \beta_{2} - 74 \beta_{3} + 18 \beta_{4} + 35 \beta_{5} + 12 \beta_{6} - 4 \beta_{7} - 11 \beta_{8} ) q^{82} + ( -444 - 21 \beta_{1} + 88 \beta_{2} + 85 \beta_{3} - 10 \beta_{4} - 16 \beta_{5} - 25 \beta_{6} + 13 \beta_{7} - 10 \beta_{8} ) q^{83} + ( 100 + 84 \beta_{1} - 35 \beta_{2} + 4 \beta_{3} - 79 \beta_{4} - 47 \beta_{5} - 26 \beta_{6} + 26 \beta_{7} + 37 \beta_{8} ) q^{84} + ( 182 + 167 \beta_{1} + 21 \beta_{2} + 65 \beta_{3} - 90 \beta_{4} + 43 \beta_{5} + 88 \beta_{6} - 38 \beta_{7} - 24 \beta_{8} ) q^{85} + ( 14 - 67 \beta_{1} + 61 \beta_{2} + 12 \beta_{3} + 47 \beta_{4} - 54 \beta_{5} + 9 \beta_{6} - 50 \beta_{7} - 33 \beta_{8} ) q^{86} + ( -132 - 30 \beta_{1} + 36 \beta_{2} - 38 \beta_{3} - 3 \beta_{4} - 18 \beta_{5} - 71 \beta_{6} + 11 \beta_{7} - 88 \beta_{8} ) q^{87} + ( 3 - 127 \beta_{1} - 20 \beta_{2} - 107 \beta_{3} + 79 \beta_{4} - 12 \beta_{5} + \beta_{6} - 22 \beta_{7} - 77 \beta_{8} ) q^{88} + ( -210 + 42 \beta_{1} - 9 \beta_{2} - 44 \beta_{3} - 18 \beta_{4} - 46 \beta_{5} - 17 \beta_{6} - 29 \beta_{7} + 13 \beta_{8} ) q^{89} + ( -158 - 5 \beta_{1} + 82 \beta_{2} + 8 \beta_{3} + 75 \beta_{4} - 48 \beta_{5} - 36 \beta_{6} + 26 \beta_{7} + 35 \beta_{8} ) q^{90} + ( 418 + 96 \beta_{1} - 46 \beta_{2} - 42 \beta_{3} - 35 \beta_{4} - 35 \beta_{5} + 31 \beta_{6} + 10 \beta_{7} + 42 \beta_{8} ) q^{92} + ( 150 - 86 \beta_{1} - 96 \beta_{2} - 68 \beta_{3} + 35 \beta_{4} - 18 \beta_{5} - 67 \beta_{6} - 27 \beta_{7} - 82 \beta_{8} ) q^{93} + ( -226 - 92 \beta_{1} + 39 \beta_{2} + 54 \beta_{3} + 61 \beta_{4} - 85 \beta_{5} + 56 \beta_{6} - 50 \beta_{7} - 27 \beta_{8} ) q^{94} + ( -206 + 140 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} - 16 \beta_{4} + 14 \beta_{5} + 37 \beta_{6} - 55 \beta_{7} + 37 \beta_{8} ) q^{95} + ( 298 + 38 \beta_{1} + 7 \beta_{2} + 47 \beta_{3} + 7 \beta_{4} - 8 \beta_{5} - 36 \beta_{6} - 28 \beta_{7} - 53 \beta_{8} ) q^{96} + ( 157 - 96 \beta_{1} + 54 \beta_{2} + 68 \beta_{3} + 87 \beta_{4} - 17 \beta_{5} - 20 \beta_{6} + 6 \beta_{7} + 8 \beta_{8} ) q^{97} + ( -33 - 26 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} + 15 \beta_{4} + 32 \beta_{5} - 153 \beta_{6} + 78 \beta_{7} - 10 \beta_{8} ) q^{98} + ( -121 - 50 \beta_{1} - 38 \beta_{2} - 70 \beta_{3} + 20 \beta_{4} + 18 \beta_{5} + 14 \beta_{6} - 14 \beta_{7} - 100 \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9} + O(q^{10})$$ $$9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9} - 147 q^{10} - 181 q^{11} + 39 q^{12} - 147 q^{14} - 218 q^{15} + 269 q^{16} - 55 q^{17} - 79 q^{18} - 161 q^{19} - 370 q^{20} - 188 q^{21} + 340 q^{22} - 204 q^{23} - 798 q^{24} + 307 q^{25} - 668 q^{27} - 344 q^{28} + 280 q^{29} + 521 q^{30} - 706 q^{31} - 680 q^{32} - 500 q^{33} - 216 q^{34} + 20 q^{35} - 909 q^{36} - 298 q^{37} - 739 q^{38} + 13 q^{40} - 1201 q^{41} - 4 q^{42} - 533 q^{43} - 355 q^{44} + 90 q^{45} + 840 q^{46} - 956 q^{47} - 132 q^{48} + 403 q^{49} + 1156 q^{50} + 470 q^{51} - 278 q^{53} + 2555 q^{54} - 250 q^{55} + 250 q^{56} + 810 q^{57} + 2877 q^{58} - 1377 q^{59} + 3157 q^{60} - 136 q^{61} + 2035 q^{62} + 944 q^{63} + 284 q^{64} + 3279 q^{66} + 931 q^{67} - 1536 q^{68} - 2050 q^{69} + 4854 q^{70} - 2046 q^{71} + 4342 q^{72} + 45 q^{73} - 1990 q^{74} + 2393 q^{75} + 3608 q^{76} - 718 q^{77} + 412 q^{79} + 787 q^{80} - 835 q^{81} + 2757 q^{82} - 3709 q^{83} + 1539 q^{84} + 2106 q^{85} - 125 q^{86} - 786 q^{87} - 636 q^{88} - 1663 q^{89} - 1280 q^{90} + 4010 q^{92} + 1186 q^{93} - 2531 q^{94} - 1614 q^{95} + 3084 q^{96} + 1087 q^{97} + 282 q^{98} - 1357 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 4 x^{8} - 46 x^{7} + 145 x^{6} + 680 x^{5} - 1501 x^{4} - 3203 x^{3} + 4784 x^{2} + 3584 x - 4096$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-6901 \nu^{8} - 12396 \nu^{7} + 501446 \nu^{6} + 728955 \nu^{5} - 11530440 \nu^{4} - 12738943 \nu^{3} + 91159455 \nu^{2} + 52058000 \nu - 149293824$$$$)/8071424$$ $$\beta_{3}$$ $$=$$ $$($$$$-3843 \nu^{8} + 11372 \nu^{7} + 195178 \nu^{6} - 384275 \nu^{5} - 3297016 \nu^{4} + 4265751 \nu^{3} + 16793257 \nu^{2} - 22791952 \nu - 6050816$$$$)/4035712$$ $$\beta_{4}$$ $$=$$ $$($$$$-595 \nu^{8} + 2737 \nu^{7} + 19422 \nu^{6} - 72705 \nu^{5} - 171803 \nu^{4} + 485407 \nu^{3} + 252488 \nu^{2} - 1604391 \nu - 646464$$$$)/504464$$ $$\beta_{5}$$ $$=$$ $$($$$$-23843 \nu^{8} + 103372 \nu^{7} + 1059978 \nu^{6} - 3803155 \nu^{5} - 14845688 \nu^{4} + 38793527 \nu^{3} + 67400873 \nu^{2} - 97179408 \nu - 92826880$$$$)/8071424$$ $$\beta_{6}$$ $$=$$ $$($$$$-32107 \nu^{8} + 90940 \nu^{7} + 1548474 \nu^{6} - 2570427 \nu^{5} - 23878504 \nu^{4} + 12527327 \nu^{3} + 111900273 \nu^{2} + 12391680 \nu - 122506496$$$$)/8071424$$ $$\beta_{7}$$ $$=$$ $$($$$$-53515 \nu^{8} + 88524 \nu^{7} + 2837370 \nu^{6} - 2210427 \nu^{5} - 46218904 \nu^{4} + 3078399 \nu^{3} + 217315329 \nu^{2} + 51740944 \nu - 202361088$$$$)/8071424$$ $$\beta_{8}$$ $$=$$ $$($$$$27975 \nu^{8} - 97156 \nu^{7} - 1304226 \nu^{6} + 3186791 \nu^{5} + 19362096 \nu^{4} - 25660427 \nu^{3} - 85614861 \nu^{2} + 34322440 \nu + 59238144$$$$)/4035712$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} + \beta_{6} + \beta_{5} + 2 \beta_{1} + 12$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{8} + 3 \beta_{6} + 2 \beta_{5} + 4 \beta_{3} - \beta_{2} + 23 \beta_{1} + 12$$ $$\nu^{4}$$ $$=$$ $$32 \beta_{8} + 37 \beta_{6} + 27 \beta_{5} - 5 \beta_{4} + 8 \beta_{3} - 8 \beta_{2} + 77 \beta_{1} + 260$$ $$\nu^{5}$$ $$=$$ $$128 \beta_{8} - 10 \beta_{7} + 166 \beta_{6} + 75 \beta_{5} - 18 \beta_{4} + 138 \beta_{3} - 45 \beta_{2} + 636 \beta_{1} + 604$$ $$\nu^{6}$$ $$=$$ $$1004 \beta_{8} - 46 \beta_{7} + 1315 \beta_{6} + 748 \beta_{5} - 299 \beta_{4} + 490 \beta_{3} - 339 \beta_{2} + 2746 \beta_{1} + 6752$$ $$\nu^{7}$$ $$=$$ $$4677 \beta_{8} - 644 \beta_{7} + 6896 \beta_{6} + 2684 \beta_{5} - 1299 \beta_{4} + 4588 \beta_{3} - 1762 \beta_{2} + 19224 \beta_{1} + 23360$$ $$\nu^{8}$$ $$=$$ $$32278 \beta_{8} - 3242 \beta_{7} + 46550 \beta_{6} + 21858 \beta_{5} - 12940 \beta_{4} + 21190 \beta_{3} - 12178 \beta_{2} + 95033 \beta_{1} + 192772$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.42835 −3.82555 −2.16135 −1.22799 0.850942 1.39012 2.72763 4.83438 5.84018
−5.42835 1.67510 21.4670 −7.70909 −9.09301 15.0250 −73.1038 −24.1941 41.8477
1.2 −4.82555 4.44352 15.2860 12.7712 −21.4425 −26.1871 −35.1589 −7.25513 −61.6281
1.3 −3.16135 7.08883 1.99415 −13.6039 −22.4103 14.3315 18.9866 23.2516 43.0068
1.4 −2.22799 −9.74867 −3.03607 8.20685 21.7199 −8.35495 24.5882 68.0366 −18.2848
1.5 −0.149058 −6.48858 −7.97778 10.2526 0.967177 29.6743 2.38162 15.1017 −1.52823
1.6 0.390115 3.60967 −7.84781 7.52136 1.40819 −19.5446 −6.18247 −13.9703 2.93420
1.7 1.72763 6.89591 −5.01528 −20.8281 11.9136 −7.56566 −22.4856 20.5536 −35.9833
1.8 3.83438 −0.279163 6.70249 −11.3710 −1.07042 −31.0623 −4.97517 −26.9221 −43.6008
1.9 4.84018 −6.19662 15.4273 −15.2399 −29.9927 −4.31620 35.9495 11.3981 −73.7636
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.a.k 9
3.b odd 2 1 1521.4.a.bh 9
13.b even 2 1 169.4.a.l yes 9
13.c even 3 2 169.4.c.l 18
13.d odd 4 2 169.4.b.g 18
13.e even 6 2 169.4.c.k 18
13.f odd 12 4 169.4.e.h 36
39.d odd 2 1 1521.4.a.bg 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.4.a.k 9 1.a even 1 1 trivial
169.4.a.l yes 9 13.b even 2 1
169.4.b.g 18 13.d odd 4 2
169.4.c.k 18 13.e even 6 2
169.4.c.l 18 13.c even 3 2
169.4.e.h 36 13.f odd 12 4
1521.4.a.bg 9 39.d odd 2 1
1521.4.a.bh 9 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{9} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(169))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$344 + 1464 T - 5898 T^{2} - 1257 T^{3} + 2310 T^{4} + 486 T^{5} - 205 T^{6} - 42 T^{7} + 5 T^{8} + T^{9}$$
$3$ $$-143717 - 375830 T + 472696 T^{2} - 97525 T^{3} - 24453 T^{4} + 7245 T^{5} + 359 T^{6} - 154 T^{7} - T^{8} + T^{9}$$
$5$ $$3059376152 - 79067576 T - 128609278 T^{2} + 1389429 T^{3} + 1945112 T^{4} + 11193 T^{5} - 12686 T^{6} - 266 T^{7} + 30 T^{8} + T^{9}$$
$7$ $$-27715644424 - 12058090668 T - 1223096892 T^{2} + 80750883 T^{3} + 15288900 T^{4} + 47784 T^{5} - 49156 T^{6} - 1023 T^{7} + 38 T^{8} + T^{9}$$
$11$ $$276199564381 - 179311762994 T + 26882546428 T^{2} + 738228615 T^{3} - 167003035 T^{4} - 5844107 T^{5} + 165265 T^{6} + 10834 T^{7} + 181 T^{8} + T^{9}$$
$13$ $$T^{9}$$
$17$ $$5572934105557 - 13634726454440 T - 8401319709328 T^{2} - 224267154861 T^{3} + 6767325927 T^{4} + 147895979 T^{5} - 1146115 T^{6} - 23436 T^{7} + 55 T^{8} + T^{9}$$
$19$ $$865058822963419 + 17958379163746 T - 7470263773658 T^{2} - 8601436031 T^{3} + 7019204857 T^{4} + 20615523 T^{5} - 2116695 T^{6} - 11292 T^{7} + 161 T^{8} + T^{9}$$
$23$ $$68005016808744392 + 1677405262433836 T - 96771175504120 T^{2} - 1742693252577 T^{3} + 41479717140 T^{4} + 357001151 T^{5} - 5407664 T^{6} - 29131 T^{7} + 204 T^{8} + T^{9}$$
$29$ $$56071014835079896 + 5762250822516012 T + 56291593140298 T^{2} - 6977746485675 T^{3} - 132290324688 T^{4} + 976035630 T^{5} + 12568120 T^{6} - 48535 T^{7} - 280 T^{8} + T^{9}$$
$31$ $$-3096277304854445656 - 65749435529097852 T + 1985161101259952 T^{2} + 25565731686363 T^{3} - 228326104668 T^{4} - 3609315492 T^{5} - 1130104 T^{6} + 148229 T^{7} + 706 T^{8} + T^{9}$$
$37$ $$-$$$$31\!\cdots\!48$$$$- 8896933236559731448 T - 45875333850181822 T^{2} + 594222222755749 T^{3} + 6293535923688 T^{4} + 7976918585 T^{5} - 87939738 T^{6} - 232098 T^{7} + 298 T^{8} + T^{9}$$
$41$ $$61873581397572169253 + 1619792110119036588 T - 1616425926085138 T^{2} - 421646173565359 T^{3} - 4652224924458 T^{4} - 14140274357 T^{5} + 56821158 T^{6} + 494135 T^{7} + 1201 T^{8} + T^{9}$$
$43$ $$-$$$$35\!\cdots\!77$$$$- 4678442324800455822 T + 70886710557000845 T^{2} + 1406432909501023 T^{3} + 5706993788145 T^{4} - 14526437912 T^{5} - 136823024 T^{6} - 153079 T^{7} + 533 T^{8} + T^{9}$$
$47$ $$11464353104027851384 + 1329362167362062820 T + 32303753105700258 T^{2} - 49762716955757 T^{3} - 6545051933164 T^{4} - 53107853183 T^{5} - 127738074 T^{6} + 136612 T^{7} + 956 T^{8} + T^{9}$$
$53$ $$-$$$$12\!\cdots\!76$$$$+$$$$24\!\cdots\!24$$$$T - 412869644476868336 T^{2} - 9184309039838179 T^{3} + 15501811028404 T^{4} + 112216880490 T^{5} - 123482118 T^{6} - 566375 T^{7} + 278 T^{8} + T^{9}$$
$59$ $$27\!\cdots\!23$$$$+$$$$10\!\cdots\!24$$$$T + 1114668162573142982 T^{2} + 2587489249146439 T^{3} - 30104186322184 T^{4} - 204479067637 T^{5} - 372100382 T^{6} + 252933 T^{7} + 1377 T^{8} + T^{9}$$
$61$ $$-$$$$27\!\cdots\!32$$$$-$$$$11\!\cdots\!08$$$$T - 13633889619625467612 T^{2} - 42822229764922651 T^{3} + 89968446968294 T^{4} + 403231012325 T^{5} - 187548706 T^{6} - 1131006 T^{7} + 136 T^{8} + T^{9}$$
$67$ $$32\!\cdots\!99$$$$+$$$$52\!\cdots\!26$$$$T + 10821001517180519169 T^{2} - 92063039319250813 T^{3} - 233825752125911 T^{4} + 495065391476 T^{5} + 890936272 T^{6} - 1141335 T^{7} - 931 T^{8} + T^{9}$$
$71$ $$44\!\cdots\!72$$$$+$$$$19\!\cdots\!12$$$$T - 49455590099883069580 T^{2} + 78182065417258095 T^{3} + 405356603728390 T^{4} - 434754292148 T^{5} - 1448605362 T^{6} + 308287 T^{7} + 2046 T^{8} + T^{9}$$
$73$ $$-$$$$31\!\cdots\!21$$$$-$$$$15\!\cdots\!50$$$$T + 17310298486778544639 T^{2} - 32424645086193109 T^{3} - 181643944920255 T^{4} + 609692221608 T^{5} + 173876306 T^{6} - 1609687 T^{7} - 45 T^{8} + T^{9}$$
$79$ $$63\!\cdots\!88$$$$+$$$$79\!\cdots\!44$$$$T + 556477632229543614 T^{2} - 21981799979683177 T^{3} - 36065298040498 T^{4} + 234613432336 T^{5} + 282452944 T^{6} - 1032469 T^{7} - 412 T^{8} + T^{9}$$
$83$ $$14\!\cdots\!61$$$$-$$$$26\!\cdots\!46$$$$T -$$$$47\!\cdots\!03$$$$T^{2} - 2443290848701223317 T^{3} - 5791016389917361 T^{4} - 6314534308340 T^{5} - 1123208654 T^{6} + 4107377 T^{7} + 3709 T^{8} + T^{9}$$
$89$ $$43\!\cdots\!23$$$$+$$$$55\!\cdots\!94$$$$T +$$$$27\!\cdots\!79$$$$T^{2} + 599122717986710443 T^{3} + 259318680361049 T^{4} - 1289685028236 T^{5} - 2134074534 T^{6} - 271263 T^{7} + 1663 T^{8} + T^{9}$$
$97$ $$20\!\cdots\!89$$$$+$$$$11\!\cdots\!26$$$$T -$$$$14\!\cdots\!87$$$$T^{2} - 1296433223643319245 T^{3} - 484271211396037 T^{4} + 3490144221316 T^{5} + 1955124078 T^{6} - 3079171 T^{7} - 1087 T^{8} + T^{9}$$