# Properties

 Label 1849.4.a.d Level $1849$ Weight $4$ Character orbit 1849.a Self dual yes Analytic conductor $109.095$ Analytic rank $0$ Dimension $10$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1849 = 43^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1849.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$109.094531601$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - x^{9} - 59 x^{8} + 42 x^{7} + 1187 x^{6} - 541 x^{5} - 9389 x^{4} + 2180 x^{3} + 22676 x^{2} - 320 x - 768$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 43) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} -\beta_{5} q^{3} + ( 4 + \beta_{2} ) q^{4} + ( 2 - \beta_{7} ) q^{5} + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{6} + ( 5 + \beta_{2} + \beta_{6} ) q^{7} + ( -4 - 3 \beta_{1} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{8} + ( 12 + \beta_{1} - \beta_{4} - \beta_{7} + \beta_{8} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} -\beta_{5} q^{3} + ( 4 + \beta_{2} ) q^{4} + ( 2 - \beta_{7} ) q^{5} + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{6} + ( 5 + \beta_{2} + \beta_{6} ) q^{7} + ( -4 - 3 \beta_{1} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{8} + ( 12 + \beta_{1} - \beta_{4} - \beta_{7} + \beta_{8} ) q^{9} + ( -3 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} ) q^{10} + ( 3 - 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{11} + ( 6 + \beta_{1} + 4 \beta_{2} + \beta_{4} - 4 \beta_{5} + \beta_{8} - \beta_{9} ) q^{12} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{13} + ( -10 - 10 \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{14} + ( -6 + 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{15} + ( 5 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - 5 \beta_{5} + \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{16} + ( 6 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - \beta_{8} ) q^{17} + ( -26 - 14 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{18} + ( -8 - 8 \beta_{1} + \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} ) q^{19} + ( 50 + 9 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{20} + ( -7 - 2 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} + 4 \beta_{6} + 5 \beta_{7} - 3 \beta_{8} ) q^{21} + ( 19 - 8 \beta_{1} - 4 \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{22} + ( 2 - \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - 9 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} ) q^{23} + ( -18 - 23 \beta_{1} + 9 \beta_{5} + 7 \beta_{6} + 8 \beta_{7} - 6 \beta_{8} + \beta_{9} ) q^{24} + ( 12 + 3 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{25} + ( -7 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + \beta_{6} + 5 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{26} + ( -10 + \beta_{1} - 4 \beta_{2} - \beta_{3} - 4 \beta_{5} - 5 \beta_{6} + \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{27} + ( 80 - 12 \beta_{1} + 15 \beta_{2} + 4 \beta_{3} - \beta_{4} - 6 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{28} + ( 7 + 11 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 5 \beta_{6} + 6 \beta_{8} - \beta_{9} ) q^{29} + ( -65 - 5 \beta_{2} + 7 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} + 14 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} ) q^{30} + ( -28 + 16 \beta_{1} + \beta_{2} - 4 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{31} + ( -37 + 5 \beta_{1} - 8 \beta_{2} + 7 \beta_{3} - 2 \beta_{4} + 10 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{32} + ( 45 + 7 \beta_{1} + 7 \beta_{2} + 10 \beta_{3} - 3 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} + 6 \beta_{8} + \beta_{9} ) q^{33} + ( 35 - 16 \beta_{1} - 5 \beta_{2} - \beta_{3} - 5 \beta_{4} + 19 \beta_{5} + \beta_{6} - 11 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{34} + ( 43 - \beta_{1} + 8 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 8 \beta_{5} - 5 \beta_{6} - 8 \beta_{7} + 2 \beta_{8} ) q^{35} + ( 112 + 39 \beta_{1} + 13 \beta_{2} + \beta_{4} + 6 \beta_{5} + 3 \beta_{6} - 13 \beta_{7} + 4 \beta_{8} + \beta_{9} ) q^{36} + ( 6 - 18 \beta_{1} - 19 \beta_{2} - \beta_{3} - 5 \beta_{4} - 14 \beta_{5} - 14 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{37} + ( 87 - 10 \beta_{1} + 15 \beta_{2} + 7 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} + 12 \beta_{6} - 4 \beta_{7} - \beta_{8} - \beta_{9} ) q^{38} + ( 70 - 28 \beta_{1} + 8 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} + 7 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} ) q^{39} + ( -121 - 50 \beta_{1} - 11 \beta_{2} + 7 \beta_{3} + 2 \beta_{4} + 23 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} ) q^{40} + ( 28 - 28 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} - 23 \beta_{5} - \beta_{6} + 10 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} ) q^{41} + ( 24 - 34 \beta_{1} + 21 \beta_{2} + 14 \beta_{3} + 4 \beta_{4} + 41 \beta_{5} + 8 \beta_{6} + 5 \beta_{7} - 5 \beta_{8} + \beta_{9} ) q^{42} + ( 18 + \beta_{1} + 11 \beta_{2} - 8 \beta_{3} + \beta_{4} - 36 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} - 7 \beta_{8} - \beta_{9} ) q^{44} + ( 110 + 20 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 18 \beta_{5} - 8 \beta_{6} - 6 \beta_{7} + 6 \beta_{8} ) q^{45} + ( 13 + 24 \beta_{1} - 7 \beta_{2} + 9 \beta_{3} + 21 \beta_{5} - 9 \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{46} + ( -24 - 41 \beta_{1} - 22 \beta_{2} + 11 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} - 25 \beta_{7} + 7 \beta_{8} - \beta_{9} ) q^{47} + ( 240 + 25 \beta_{1} + 34 \beta_{2} - 4 \beta_{3} + \beta_{4} - 10 \beta_{5} - \beta_{6} - 17 \beta_{7} + 4 \beta_{8} + \beta_{9} ) q^{48} + ( 3 - 9 \beta_{1} + 13 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 16 \beta_{5} + 2 \beta_{6} + 7 \beta_{7} - 2 \beta_{8} ) q^{49} + ( -22 + 10 \beta_{1} - 18 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 49 \beta_{5} - 4 \beta_{6} - 7 \beta_{7} - 7 \beta_{8} + \beta_{9} ) q^{50} + ( 72 + 72 \beta_{1} + 5 \beta_{2} + 13 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 15 \beta_{7} + 8 \beta_{8} + \beta_{9} ) q^{51} + ( 53 - 40 \beta_{1} + 29 \beta_{2} + 11 \beta_{3} + 3 \beta_{4} - 18 \beta_{5} + 4 \beta_{6} + 5 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{52} + ( -75 - 31 \beta_{1} - 12 \beta_{2} - 3 \beta_{3} + 7 \beta_{4} - \beta_{5} + 7 \beta_{6} + 3 \beta_{7} + 4 \beta_{8} ) q^{53} + ( 31 + 42 \beta_{1} - 15 \beta_{2} + 7 \beta_{3} - \beta_{4} + 34 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} + 6 \beta_{9} ) q^{54} + ( 110 - 17 \beta_{1} + 19 \beta_{2} - 18 \beta_{3} - 5 \beta_{4} - 29 \beta_{5} + 3 \beta_{6} + 20 \beta_{7} - 5 \beta_{8} + 6 \beta_{9} ) q^{55} + ( 84 - 120 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} - 4 \beta_{4} - 25 \beta_{5} + 12 \beta_{6} + 13 \beta_{7} - 7 \beta_{8} + \beta_{9} ) q^{56} + ( 93 - \beta_{1} + 41 \beta_{2} + 13 \beta_{3} + 4 \beta_{4} + 32 \beta_{5} + 6 \beta_{6} - 4 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} ) q^{57} + ( -120 + 13 \beta_{1} - 44 \beta_{2} - 5 \beta_{4} + 40 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} ) q^{58} + ( -174 - 2 \beta_{1} - 55 \beta_{2} - 10 \beta_{3} - 10 \beta_{4} - 35 \beta_{5} + 11 \beta_{6} - 25 \beta_{7} - 8 \beta_{8} + 6 \beta_{9} ) q^{59} + ( 66 + 102 \beta_{1} + 47 \beta_{2} - 4 \beta_{3} + 9 \beta_{4} - 84 \beta_{5} - 5 \beta_{6} + 3 \beta_{7} + 10 \beta_{8} - \beta_{9} ) q^{60} + ( -62 + \beta_{1} - 30 \beta_{2} + \beta_{3} + 10 \beta_{4} - 41 \beta_{5} - 7 \beta_{6} - 2 \beta_{7} + 7 \beta_{8} + 5 \beta_{9} ) q^{61} + ( -125 + 11 \beta_{1} - 28 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 72 \beta_{5} + 7 \beta_{6} + 20 \beta_{7} - 7 \beta_{8} + 12 \beta_{9} ) q^{62} + ( 233 + 93 \beta_{1} + 38 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} + 45 \beta_{5} + 9 \beta_{6} + 22 \beta_{7} - 8 \beta_{8} + 4 \beta_{9} ) q^{63} + ( -105 + 45 \beta_{1} + 2 \beta_{2} - 11 \beta_{3} - 5 \beta_{4} - 61 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} + 5 \beta_{8} + \beta_{9} ) q^{64} + ( -119 + 45 \beta_{1} + 5 \beta_{2} + 17 \beta_{3} + 12 \beta_{4} - 18 \beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{65} + ( -185 - 112 \beta_{1} - 32 \beta_{2} + \beta_{3} + 2 \beta_{4} - 76 \beta_{5} - \beta_{6} + 30 \beta_{7} - 10 \beta_{8} - \beta_{9} ) q^{66} + ( 103 - 46 \beta_{1} + 4 \beta_{2} + 5 \beta_{3} + 46 \beta_{5} - 13 \beta_{6} - 3 \beta_{7} + 21 \beta_{8} - 6 \beta_{9} ) q^{67} + ( 105 - 4 \beta_{1} - 17 \beta_{2} - 15 \beta_{3} - 6 \beta_{5} - 3 \beta_{6} - 20 \beta_{7} - 5 \beta_{9} ) q^{68} + ( 333 + 41 \beta_{1} - 13 \beta_{2} - 4 \beta_{3} - 16 \beta_{4} - 25 \beta_{5} + \beta_{6} - 4 \beta_{7} + 4 \beta_{8} + 7 \beta_{9} ) q^{69} + ( -14 - 101 \beta_{1} - 16 \beta_{2} - 10 \beta_{3} - 2 \beta_{4} - 7 \beta_{5} + 12 \beta_{6} + 29 \beta_{7} - 15 \beta_{8} + 5 \beta_{9} ) q^{70} + ( 169 - 118 \beta_{1} + 28 \beta_{2} + 5 \beta_{3} - \beta_{4} - 2 \beta_{5} - 9 \beta_{6} - 13 \beta_{7} - 4 \beta_{8} + 7 \beta_{9} ) q^{71} + ( -375 - 85 \beta_{1} - 17 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} + 36 \beta_{5} + 21 \beta_{6} + 40 \beta_{7} - 23 \beta_{8} + 6 \beta_{9} ) q^{72} + ( -155 + 59 \beta_{1} - 28 \beta_{2} + 3 \beta_{3} - 12 \beta_{4} - 39 \beta_{5} - 7 \beta_{6} + 11 \beta_{7} + 9 \beta_{8} - 11 \beta_{9} ) q^{73} + ( 219 + 31 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} - 21 \beta_{4} - 16 \beta_{5} + 4 \beta_{6} - 45 \beta_{7} + 14 \beta_{8} - 4 \beta_{9} ) q^{74} + ( -19 + 184 \beta_{1} + 27 \beta_{2} + 3 \beta_{3} + 16 \beta_{4} - 10 \beta_{5} + 16 \beta_{6} + 34 \beta_{7} - 3 \beta_{8} + 6 \beta_{9} ) q^{75} + ( -18 - 143 \beta_{1} + 32 \beta_{2} - 2 \beta_{3} - \beta_{4} - 101 \beta_{5} + 18 \beta_{6} - 5 \beta_{7} - 18 \beta_{8} - 6 \beta_{9} ) q^{76} + ( -112 - 92 \beta_{1} + 53 \beta_{2} - 3 \beta_{3} + 21 \beta_{4} + 30 \beta_{5} - 2 \beta_{6} + 11 \beta_{7} + 4 \beta_{8} - 5 \beta_{9} ) q^{77} + ( 311 - 151 \beta_{1} + 28 \beta_{2} - 11 \beta_{3} - 11 \beta_{4} + 36 \beta_{5} + 20 \beta_{6} - 17 \beta_{7} - 12 \beta_{8} - 2 \beta_{9} ) q^{78} + ( 127 + 58 \beta_{1} - 4 \beta_{2} - 15 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} + 11 \beta_{6} - 15 \beta_{7} - 16 \beta_{8} - 7 \beta_{9} ) q^{79} + ( 266 + 130 \beta_{1} + 43 \beta_{2} + 4 \beta_{3} + 27 \beta_{4} - 92 \beta_{5} - 5 \beta_{6} - \beta_{7} + 4 \beta_{8} + 5 \beta_{9} ) q^{80} + ( -92 + 60 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} + 13 \beta_{4} + 24 \beta_{5} - 18 \beta_{6} - 7 \beta_{7} - 11 \beta_{8} + 7 \beta_{9} ) q^{81} + ( 356 - 47 \beta_{1} + 39 \beta_{2} + 18 \beta_{3} + 7 \beta_{4} + 14 \beta_{5} - 17 \beta_{6} - 13 \beta_{7} + 16 \beta_{8} + \beta_{9} ) q^{82} + ( -4 - 101 \beta_{1} + 25 \beta_{2} + 4 \beta_{3} - 11 \beta_{4} + 58 \beta_{5} - \beta_{6} - 16 \beta_{7} + 21 \beta_{8} ) q^{83} + ( 335 - 103 \beta_{1} + 58 \beta_{2} + 3 \beta_{3} + 8 \beta_{4} - 117 \beta_{5} - 11 \beta_{6} - 19 \beta_{7} + 19 \beta_{8} - 18 \beta_{9} ) q^{84} + ( 152 - 111 \beta_{1} + 13 \beta_{2} - 22 \beta_{3} - 17 \beta_{4} + 59 \beta_{5} - 19 \beta_{6} - 25 \beta_{7} + 15 \beta_{8} - 8 \beta_{9} ) q^{85} + ( 121 + 143 \beta_{1} + 5 \beta_{2} + \beta_{3} - 8 \beta_{4} - 43 \beta_{5} - 26 \beta_{6} - 27 \beta_{7} + 21 \beta_{8} + 5 \beta_{9} ) q^{87} + ( -222 - 108 \beta_{1} + 32 \beta_{2} + 16 \beta_{3} + 16 \beta_{4} + 89 \beta_{5} + 6 \beta_{6} + 23 \beta_{7} - 21 \beta_{8} + \beta_{9} ) q^{88} + ( 191 + 110 \beta_{1} - 29 \beta_{2} - 25 \beta_{3} - 7 \beta_{4} - 20 \beta_{5} + 14 \beta_{6} - 11 \beta_{7} - 12 \beta_{8} + \beta_{9} ) q^{89} + ( -56 - 246 \beta_{1} - 60 \beta_{2} - 32 \beta_{3} - 14 \beta_{4} + 30 \beta_{5} + 14 \beta_{6} + 24 \beta_{7} - 30 \beta_{8} + 8 \beta_{9} ) q^{90} + ( 332 - 48 \beta_{1} + 22 \beta_{2} + 14 \beta_{3} - 2 \beta_{4} - 41 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} - 6 \beta_{8} - 6 \beta_{9} ) q^{91} + ( -277 + 35 \beta_{1} + 8 \beta_{2} - 7 \beta_{3} + 14 \beta_{4} - 35 \beta_{5} - 6 \beta_{6} - 12 \beta_{7} + 4 \beta_{8} ) q^{92} + ( 154 + 259 \beta_{1} + 53 \beta_{2} + 18 \beta_{3} + 23 \beta_{4} + 121 \beta_{5} - 7 \beta_{6} - 23 \beta_{7} + 29 \beta_{8} - 6 \beta_{9} ) q^{93} + ( 440 + 139 \beta_{1} - \beta_{2} - 14 \beta_{3} - 13 \beta_{4} - 79 \beta_{5} - 6 \beta_{6} + \beta_{7} - 4 \beta_{8} - 6 \beta_{9} ) q^{94} + ( -20 - 50 \beta_{1} + 21 \beta_{2} - 16 \beta_{3} + 5 \beta_{4} - 66 \beta_{5} + 5 \beta_{6} + 33 \beta_{7} - 17 \beta_{8} + 11 \beta_{9} ) q^{95} + ( -331 - 320 \beta_{1} - 91 \beta_{2} - 13 \beta_{3} - 19 \beta_{4} + 4 \beta_{5} - 18 \beta_{6} - 15 \beta_{7} - 16 \beta_{8} - 2 \beta_{9} ) q^{96} + ( -49 - 15 \beta_{1} - 11 \beta_{2} - 9 \beta_{3} + 17 \beta_{4} + 18 \beta_{5} + 24 \beta_{6} + 9 \beta_{7} - 32 \beta_{8} - 10 \beta_{9} ) q^{97} + ( 1 - 117 \beta_{1} + 45 \beta_{2} + 25 \beta_{3} + 9 \beta_{4} - 70 \beta_{5} + 10 \beta_{6} - 5 \beta_{7} + 4 \beta_{8} - 10 \beta_{9} ) q^{98} + ( 318 - 176 \beta_{1} - 44 \beta_{2} - 7 \beta_{3} + 3 \beta_{4} - 79 \beta_{5} - 45 \beta_{6} - 42 \beta_{7} + 30 \beta_{8} - 5 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - q^{2} + 5q^{3} + 39q^{4} + 19q^{5} - 15q^{6} + 51q^{7} - 36q^{8} + 117q^{9} + O(q^{10})$$ $$10q - q^{2} + 5q^{3} + 39q^{4} + 19q^{5} - 15q^{6} + 51q^{7} - 36q^{8} + 117q^{9} - 27q^{10} + 27q^{11} + 72q^{12} + 15q^{13} - 96q^{14} - 65q^{15} + 67q^{16} + 82q^{17} - 247q^{18} - 78q^{19} + 495q^{20} - 9q^{21} + 190q^{22} + 61q^{23} - 202q^{24} + 151q^{25} + 21q^{26} - 97q^{27} + 794q^{28} + 53q^{29} - 627q^{30} - 253q^{31} - 399q^{32} + 424q^{33} + 231q^{34} + 355q^{35} + 1092q^{36} + 129q^{37} + 854q^{38} + 691q^{39} - 1345q^{40} + 391q^{41} + 31q^{42} + 377q^{44} + 944q^{45} + 40q^{46} - 334q^{47} + 2401q^{48} + 115q^{49} - 424q^{50} + 795q^{51} + 564q^{52} - 773q^{53} + 182q^{54} + 1242q^{55} + 923q^{56} + 765q^{57} - 1328q^{58} - 1483q^{59} + 1075q^{60} - 437q^{61} - 1509q^{62} + 2222q^{63} - 738q^{64} - 1063q^{65} - 1483q^{66} + 642q^{67} + 1052q^{68} + 3503q^{69} - 85q^{70} + 1545q^{71} - 3834q^{72} - 1292q^{73} + 2232q^{74} + 82q^{75} + 252q^{76} - 1448q^{77} + 2822q^{78} + 1405q^{79} + 3157q^{80} - 974q^{81} + 3304q^{82} - 543q^{83} + 3652q^{84} + 973q^{85} + 1409q^{87} - 2686q^{88} + 2196q^{89} - 742q^{90} + 3513q^{91} - 2629q^{92} + 983q^{93} + 4939q^{94} + 149q^{95} - 3540q^{96} - 425q^{97} + 213q^{98} + 3181q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} - 59 x^{8} + 42 x^{7} + 1187 x^{6} - 541 x^{5} - 9389 x^{4} + 2180 x^{3} + 22676 x^{2} - 320 x - 768$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 12$$ $$\beta_{3}$$ $$=$$ $$($$$$225 \nu^{9} + 1607 \nu^{8} - 30915 \nu^{7} - 48526 \nu^{6} + 948499 \nu^{5} + 145275 \nu^{4} - 8738005 \nu^{3} + 2790076 \nu^{2} + 16689012 \nu - 1568544$$$$)/704512$$ $$\beta_{4}$$ $$=$$ $$($$$$345 \nu^{9} + 8335 \nu^{8} - 47403 \nu^{7} - 385566 \nu^{6} + 1554171 \nu^{5} + 5330467 \nu^{4} - 16950189 \nu^{3} - 23679268 \nu^{2} + 48345556 \nu + 21830112$$$$)/704512$$ $$\beta_{5}$$ $$=$$ $$($$$$-585 \nu^{9} + 225 \nu^{8} + 36347 \nu^{7} - 19138 \nu^{6} - 762059 \nu^{5} + 502925 \nu^{4} + 6140765 \nu^{3} - 3872540 \nu^{2} - 14347924 \nu + 1823776$$$$)/704512$$ $$\beta_{6}$$ $$=$$ $$($$$$-827 \nu^{9} - 3069 \nu^{8} + 51985 \nu^{7} + 139514 \nu^{6} - 1039745 \nu^{5} - 1620089 \nu^{4} + 6893047 \nu^{3} + 2149420 \nu^{2} - 5101372 \nu + 10152032$$$$)/704512$$ $$\beta_{7}$$ $$=$$ $$($$$$-1749 \nu^{9} + 3213 \nu^{8} + 95007 \nu^{7} - 129306 \nu^{6} - 1729711 \nu^{5} + 1424585 \nu^{4} + 12258905 \nu^{3} - 3579884 \nu^{2} - 26340356 \nu + 18848$$$$)/352256$$ $$\beta_{8}$$ $$=$$ $$($$$$-4325 \nu^{9} + 3357 \nu^{8} + 241999 \nu^{7} - 119098 \nu^{6} - 4499167 \nu^{5} + 1229081 \nu^{4} + 32115369 \nu^{3} - 5010348 \nu^{2} - 71167812 \nu + 7371680$$$$)/704512$$ $$\beta_{9}$$ $$=$$ $$($$$$141 \nu^{9} - 213 \nu^{8} - 9191 \nu^{7} + 10586 \nu^{6} + 207911 \nu^{5} - 170401 \nu^{4} - 1838177 \nu^{3} + 949964 \nu^{2} + 4671652 \nu - 704672$$$$)/22016$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 12$$ $$\nu^{3}$$ $$=$$ $$\beta_{8} - \beta_{7} - \beta_{6} + 19 \beta_{1} + 4$$ $$\nu^{4}$$ $$=$$ $$-\beta_{9} + 2 \beta_{8} - 3 \beta_{7} + \beta_{6} - 5 \beta_{5} + \beta_{4} + \beta_{3} + 26 \beta_{2} + 3 \beta_{1} + 229$$ $$\nu^{5}$$ $$=$$ $$\beta_{9} + 33 \beta_{8} - 31 \beta_{7} - 30 \beta_{6} - 10 \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 8 \beta_{2} + 411 \beta_{1} + 165$$ $$\nu^{6}$$ $$=$$ $$-39 \beta_{9} + 85 \beta_{8} - 114 \beta_{7} + 32 \beta_{6} - 261 \beta_{5} + 35 \beta_{4} + 29 \beta_{3} + 658 \beta_{2} + 165 \beta_{1} + 4959$$ $$\nu^{7}$$ $$=$$ $$26 \beta_{9} + 941 \beta_{8} - 858 \beta_{7} - 797 \beta_{6} - 573 \beta_{5} + 89 \beta_{4} - 328 \beta_{3} + 423 \beta_{2} + 9604 \beta_{1} + 5588$$ $$\nu^{8}$$ $$=$$ $$-1214 \beta_{9} + 2809 \beta_{8} - 3510 \beta_{7} + 731 \beta_{6} - 9313 \beta_{5} + 1071 \beta_{4} + 602 \beta_{3} + 16880 \beta_{2} + 6767 \beta_{1} + 116228$$ $$\nu^{9}$$ $$=$$ $$262 \beta_{9} + 25994 \beta_{8} - 23623 \beta_{7} - 20842 \beta_{6} - 23121 \beta_{5} + 3051 \beta_{4} - 11118 \beta_{3} + 16559 \beta_{2} + 236018 \beta_{1} + 177264$$

## 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
 5.31336 4.23473 3.55840 2.14781 0.190911 −0.179767 −1.92278 −3.59365 −3.82341 −4.92559
−5.31336 8.05642 20.2317 15.9193 −42.8066 16.6323 −64.9916 37.9059 −84.5852
1.2 −4.23473 −8.85858 9.93292 5.96488 37.5137 22.5055 −8.18537 51.4744 −25.2597
1.3 −3.55840 −0.389088 4.66219 −1.72780 1.38453 −21.1861 11.8773 −26.8486 6.14819
1.4 −2.14781 6.84883 −3.38692 −17.1363 −14.7100 22.9786 24.4569 19.9064 36.8056
1.5 −0.190911 1.43836 −7.96355 16.3462 −0.274600 6.23994 3.04762 −24.9311 −3.12068
1.6 0.179767 −6.02248 −7.96768 −10.9703 −1.08264 8.78367 −2.87047 9.27026 −1.97211
1.7 1.92278 8.37832 −4.30290 0.0702257 16.1097 −23.4425 −23.6558 43.1963 0.135029
1.8 3.59365 −1.67792 4.91431 −9.72181 −6.02986 −15.7603 −11.0889 −24.1846 −34.9368
1.9 3.82341 −7.76517 6.61843 18.1323 −29.6894 5.19796 −5.28231 33.2979 69.3271
1.10 4.92559 4.99131 16.2615 2.12330 24.5852 29.0509 40.6926 −2.08684 10.4585
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$43$$ $$1$$

## 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 1849.4.a.d 10
43.b odd 2 1 1849.4.a.f 10
43.c even 3 2 43.4.c.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.4.c.a 20 43.c even 3 2
1849.4.a.d 10 1.a even 1 1 trivial
1849.4.a.f 10 43.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1849))$$.