# Properties

 Label 847.2.a.m Level $847$ Weight $2$ Character orbit 847.a Self dual yes Analytic conductor $6.763$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 847.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.76332905120$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.7674048.1 Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2$$ x^6 - 2*x^5 - 5*x^4 + 8*x^3 + 7*x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + \beta_{5} q^{3} + (\beta_{2} - \beta_1 + 1) q^{4} + ( - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{6} - q^{7} + (\beta_{3} - 2 \beta_{2} - 2) q^{8} + (\beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + b5 * q^3 + (b2 - b1 + 1) * q^4 + (-b4 + b2 + b1 - 1) * q^5 + (-b5 + b4 - b3 - b1 + 1) * q^6 - q^7 + (b3 - 2*b2 - 2) * q^8 + (b4 - b3 - b2 - 2*b1 + 2) * q^9 $$q + (\beta_1 - 1) q^{2} + \beta_{5} q^{3} + (\beta_{2} - \beta_1 + 1) q^{4} + ( - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{6} - q^{7} + (\beta_{3} - 2 \beta_{2} - 2) q^{8} + (\beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{9} + ( - \beta_{5} - \beta_{2} + 1) q^{10} + ( - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{12} + ( - \beta_{5} + \beta_{4} - \beta_{2} - 1) q^{13} + ( - \beta_1 + 1) q^{14} + ( - 2 \beta_{5} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{15} + (\beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{16} + (\beta_{3} + \beta_{2} + \beta_1 - 4) q^{17} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 4) q^{18} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_1 - 1) q^{19} + (\beta_{5} + \beta_{4} - \beta_{2} - \beta_1 + 1) q^{20} - \beta_{5} q^{21} + (3 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{23} + (\beta_{4} - 4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 4) q^{24} + ( - 2 \beta_{5} - 4 \beta_{2}) q^{25} + (2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{26} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{27} + ( - \beta_{2} + \beta_1 - 1) q^{28} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{29} + (2 \beta_{5} - \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{30} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{31} + (\beta_{5} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{32} + (\beta_{4} + \beta_{2} - 3 \beta_1 + 5) q^{34} + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{35} + ( - 2 \beta_{5} - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 3) q^{36} + (2 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{37} + ( - \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 4) q^{38} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{39} + (2 \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{40} + ( - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 4) q^{41} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{42} + (\beta_{5} + 3 \beta_{4} - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{43} + (\beta_{5} - \beta_{4} + \beta_{3} + 4 \beta_{2} + \beta_1 - 6) q^{45} + (3 \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} - 1) q^{46} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 3) q^{47} + (\beta_{5} + 3 \beta_{3} - 7 \beta_{2} + 4 \beta_1 - 5) q^{48} + q^{49} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 2) q^{50} + ( - 5 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{51} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{52} + ( - 3 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{53} + ( - \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 6) q^{54} + ( - \beta_{3} + 2 \beta_{2} + 2) q^{56} + ( - \beta_{5} - 3 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{57} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 1) q^{58} + ( - 3 \beta_{5} + 3 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 3) q^{59} + (2 \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_1 + 5) q^{60} + (2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 6 \beta_1 + 4) q^{61} + ( - 3 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 4) q^{62} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{63} + ( - 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3) q^{64} + (3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 3) q^{65} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 4 \beta_1 - 1) q^{67} + (\beta_{5} - 5 \beta_{2} + 4 \beta_1 - 3) q^{68} + (2 \beta_{5} - 3 \beta_{3} - \beta_{2} + 4 \beta_1 - 3) q^{69} + (\beta_{5} + \beta_{2} - 1) q^{70} + (2 \beta_{5} - \beta_{2} + \beta_1 + 4) q^{71} + ( - 2 \beta_{4} + 6 \beta_{3} - 6 \beta_{2} + 7 \beta_1 - 5) q^{72} + (3 \beta_{5} + 2 \beta_{4} + \beta_{3} + 4 \beta_{2} - \beta_1 - 1) q^{73} + ( - \beta_{5} + 3 \beta_{4} - 3 \beta_{2} + 4 \beta_1 - 9) q^{74} + (4 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 6 \beta_{2} - 2) q^{75} + (3 \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_1 + 7) q^{76} + ( - 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 6) q^{78} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 6) q^{79} + ( - 3 \beta_{5} - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{80} + (\beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{81} + ( - \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{82} + ( - \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 3 \beta_1 - 3) q^{83} + (2 \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{84} + (\beta_{5} + 5 \beta_{4} - \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 6) q^{85} + (2 \beta_{5} - \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 5) q^{86} + ( - \beta_{5} - \beta_{3} + 3 \beta_1 - 5) q^{87} + ( - \beta_{5} - 3 \beta_{4} - \beta_{3} - 4 \beta_{2} - \beta_1) q^{89} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 4) q^{90} + (\beta_{5} - \beta_{4} + \beta_{2} + 1) q^{91} + ( - 4 \beta_{5} - 4 \beta_{3} + 3 \beta_{2} + \beta_1) q^{92} + ( - 2 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} - \beta_{2} + 3 \beta_1 - 10) q^{93} + (\beta_{5} + 5 \beta_{3} - \beta_1 + 7) q^{94} + (3 \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_{2} + 5) q^{95} + ( - \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - 9 \beta_1 + 13) q^{96} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{97} + (\beta_1 - 1) q^{98}+O(q^{100})$$ q + (b1 - 1) * q^2 + b5 * q^3 + (b2 - b1 + 1) * q^4 + (-b4 + b2 + b1 - 1) * q^5 + (-b5 + b4 - b3 - b1 + 1) * q^6 - q^7 + (b3 - 2*b2 - 2) * q^8 + (b4 - b3 - b2 - 2*b1 + 2) * q^9 + (-b5 - b2 + 1) * q^10 + (-2*b4 + 3*b3 - b2 + 2*b1 - 3) * q^12 + (-b5 + b4 - b2 - 1) * q^13 + (-b1 + 1) * q^14 + (-2*b5 + 2*b3 - b2 - b1) * q^15 + (b4 - 3*b3 + b2 - 2*b1 + 2) * q^16 + (b3 + b2 + b1 - 4) * q^17 + (b5 - b4 + b3 - b2 + b1 - 4) * q^18 + (-b5 - 2*b4 + b3 - b1 - 1) * q^19 + (b5 + b4 - b2 - b1 + 1) * q^20 - b5 * q^21 + (3*b4 - b3 - b2 - 2*b1 + 1) * q^23 + (b4 - 4*b3 + 4*b2 - 2*b1 + 4) * q^24 + (-2*b5 - 4*b2) * q^25 + (2*b5 - b4 + b3 + 2*b2 - b1 + 2) * q^26 + (-b3 - b2 + 2*b1 - 1) * q^27 + (-b2 + b1 - 1) * q^28 + (-b5 + b4 - b3 - b2 - b1 - 2) * q^29 + (2*b5 - b3 + 2*b2 + b1 - 3) * q^30 + (-2*b5 - 2*b4 - b3 + b2 - 1) * q^31 + (b5 - 3*b4 + 3*b3 - b2 + 3*b1 - 2) * q^32 + (b4 + b2 - 3*b1 + 5) * q^34 + (b4 - b2 - b1 + 1) * q^35 + (-2*b5 - 2*b3 + 4*b2 - 2*b1 + 3) * q^36 + (2*b5 + b4 + b3 + 2*b2 - 3*b1 + 4) * q^37 + (-b5 - 2*b3 - 2*b2 - 4) * q^38 + (-2*b3 + 2*b2 + 2*b1 - 4) * q^39 + (2*b5 + b4 - b3 + 3*b2 - b1 - 2) * q^40 + (-b4 + b3 + b2 - b1 - 4) * q^41 + (b5 - b4 + b3 + b1 - 1) * q^42 + (b5 + 3*b4 - 2*b3 + b2 - 3*b1 + 2) * q^43 + (b5 - b4 + b3 + 4*b2 + b1 - 6) * q^45 + (3*b5 - b4 + 3*b3 + b2 - 1) * q^46 + (b5 + 2*b4 - b3 + 3*b2 + 2*b1 - 3) * q^47 + (b5 + 3*b3 - 7*b2 + 4*b1 - 5) * q^48 + q^49 + (2*b5 - 2*b4 - 2*b3 + 4*b2 - 2*b1 + 2) * q^50 + (-5*b5 - b4 + b3 + b2 - 1) * q^51 + (-b5 + b4 - 2*b3 - b2 + 2*b1 - 3) * q^52 + (-3*b4 + b3 + b2 + 2*b1) * q^53 + (-b4 + 2*b2 - 2*b1 + 6) * q^54 + (-b3 + 2*b2 + 2) * q^56 + (-b5 - 3*b4 + 4*b3 + 3*b2 + b1 - 4) * q^57 + (2*b5 - 2*b4 + 2*b3 - 2*b1 + 1) * q^58 + (-3*b5 + 3*b3 - 3*b2 + 4*b1 - 3) * q^59 + (2*b5 + b4 - 3*b3 - b1 + 5) * q^60 + (2*b5 + b4 + 2*b3 - 4*b2 - 6*b1 + 4) * q^61 + (-3*b4 + 2*b3 - 4*b2 + 2*b1 - 4) * q^62 + (-b4 + b3 + b2 + 2*b1 - 2) * q^63 + (-4*b5 + 2*b4 - 2*b3 + 2*b2 + 3) * q^64 + (3*b5 + b4 - 2*b3 + 3*b2 - 3) * q^65 + (-2*b5 + b4 + b3 - b2 + 4*b1 - 1) * q^67 + (b5 - 5*b2 + 4*b1 - 3) * q^68 + (2*b5 - 3*b3 - b2 + 4*b1 - 3) * q^69 + (b5 + b2 - 1) * q^70 + (2*b5 - b2 + b1 + 4) * q^71 + (-2*b4 + 6*b3 - 6*b2 + 7*b1 - 5) * q^72 + (3*b5 + 2*b4 + b3 + 4*b2 - b1 - 1) * q^73 + (-b5 + 3*b4 - 3*b2 + 4*b1 - 9) * q^74 + (4*b5 + 2*b4 - 6*b3 + 6*b2 - 2) * q^75 + (3*b5 + b4 - b3 - 3*b1 + 7) * q^76 + (-2*b4 + 4*b3 - 2*b2 - 2*b1 + 6) * q^78 + (b5 - b4 - 2*b3 - 3*b2 - 3*b1 + 6) * q^79 + (-3*b5 - b4 + 3*b3 - 2*b2 + b1 - 2) * q^80 + (b4 - b3 + 2*b2 + 3*b1 - 2) * q^81 + (-b5 + b4 - b3 - 2*b2 - 3*b1) * q^82 + (-b5 + 2*b4 - 3*b3 - 3*b1 - 3) * q^83 + (2*b4 - 3*b3 + b2 - 2*b1 + 3) * q^84 + (b5 + 5*b4 - b3 - 5*b2 - 5*b1 + 6) * q^85 + (2*b5 - b4 + 5*b3 - 3*b2 + 2*b1 - 5) * q^86 + (-b5 - b3 + 3*b1 - 5) * q^87 + (-b5 - 3*b4 - b3 - 4*b2 - b1) * q^89 + (-2*b5 + 2*b4 + b3 - 3*b2 - 3*b1 + 4) * q^90 + (b5 - b4 + b2 + 1) * q^91 + (-4*b5 - 4*b3 + 3*b2 + b1) * q^92 + (-2*b5 - 2*b4 + 6*b3 - b2 + 3*b1 - 10) * q^93 + (b5 + 5*b3 - b1 + 7) * q^94 + (3*b5 + b4 - 3*b3 - b2 + 5) * q^95 + (-b5 + 2*b4 - 3*b3 + 6*b2 - 9*b1 + 13) * q^96 + (2*b5 + b4 - 2*b3 + b2 + 3*b1 - 1) * q^97 + (b1 - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10})$$ 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9 $$6 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{7} - 12 q^{8} + 8 q^{9} + 8 q^{10} - 14 q^{12} - 4 q^{13} + 4 q^{14} + 2 q^{15} + 8 q^{16} - 22 q^{17} - 24 q^{18} - 6 q^{19} + 2 q^{20} + 2 q^{21} + 2 q^{23} + 20 q^{24} + 4 q^{25} + 6 q^{26} - 2 q^{27} - 4 q^{28} - 12 q^{29} - 20 q^{30} - 2 q^{31} - 8 q^{32} + 24 q^{34} + 4 q^{35} + 18 q^{36} + 14 q^{37} - 22 q^{38} - 20 q^{39} - 18 q^{40} - 26 q^{41} - 6 q^{42} + 4 q^{43} - 36 q^{45} - 12 q^{46} - 16 q^{47} - 24 q^{48} + 6 q^{49} + 4 q^{50} + 4 q^{51} - 12 q^{52} + 4 q^{53} + 32 q^{54} + 12 q^{56} - 20 q^{57} - 2 q^{58} - 4 q^{59} + 24 q^{60} + 8 q^{61} - 20 q^{62} - 8 q^{63} + 26 q^{64} - 24 q^{65} + 6 q^{67} - 12 q^{68} - 14 q^{69} - 8 q^{70} + 22 q^{71} - 16 q^{72} - 14 q^{73} - 44 q^{74} - 20 q^{75} + 30 q^{76} + 32 q^{78} + 28 q^{79} - 4 q^{80} - 6 q^{81} - 4 q^{82} - 22 q^{83} + 14 q^{84} + 24 q^{85} - 30 q^{86} - 22 q^{87} + 22 q^{90} + 4 q^{91} + 10 q^{92} - 50 q^{93} + 38 q^{94} + 24 q^{95} + 62 q^{96} - 4 q^{97} - 4 q^{98}+O(q^{100})$$ 6 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^7 - 12 * q^8 + 8 * q^9 + 8 * q^10 - 14 * q^12 - 4 * q^13 + 4 * q^14 + 2 * q^15 + 8 * q^16 - 22 * q^17 - 24 * q^18 - 6 * q^19 + 2 * q^20 + 2 * q^21 + 2 * q^23 + 20 * q^24 + 4 * q^25 + 6 * q^26 - 2 * q^27 - 4 * q^28 - 12 * q^29 - 20 * q^30 - 2 * q^31 - 8 * q^32 + 24 * q^34 + 4 * q^35 + 18 * q^36 + 14 * q^37 - 22 * q^38 - 20 * q^39 - 18 * q^40 - 26 * q^41 - 6 * q^42 + 4 * q^43 - 36 * q^45 - 12 * q^46 - 16 * q^47 - 24 * q^48 + 6 * q^49 + 4 * q^50 + 4 * q^51 - 12 * q^52 + 4 * q^53 + 32 * q^54 + 12 * q^56 - 20 * q^57 - 2 * q^58 - 4 * q^59 + 24 * q^60 + 8 * q^61 - 20 * q^62 - 8 * q^63 + 26 * q^64 - 24 * q^65 + 6 * q^67 - 12 * q^68 - 14 * q^69 - 8 * q^70 + 22 * q^71 - 16 * q^72 - 14 * q^73 - 44 * q^74 - 20 * q^75 + 30 * q^76 + 32 * q^78 + 28 * q^79 - 4 * q^80 - 6 * q^81 - 4 * q^82 - 22 * q^83 + 14 * q^84 + 24 * q^85 - 30 * q^86 - 22 * q^87 + 22 * q^90 + 4 * q^91 + 10 * q^92 - 50 * q^93 + 38 * q^94 + 24 * q^95 + 62 * q^96 - 4 * q^97 - 4 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu + 1$$ v^3 - v^2 - 3*v + 1 $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2$$ v^4 - v^3 - 4*v^2 + 2*v + 2 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 4\nu - 2$$ v^5 - 2*v^4 - 4*v^3 + 6*v^2 + 4*v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 1$$ b3 + b2 + 4*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 5\beta_{2} + 6\beta _1 + 7$$ b4 + b3 + 5*b2 + 6*b1 + 7 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 2\beta_{4} + 6\beta_{3} + 8\beta_{2} + 18\beta _1 + 8$$ b5 + 2*b4 + 6*b3 + 8*b2 + 18*b1 + 8

## 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
 −1.70320 −1.10939 −0.276564 0.879640 1.82356 2.38595
−2.70320 −2.80719 5.30727 −0.445072 7.58839 −1.00000 −8.94020 4.88032 1.20312
1.2 −2.10939 1.69851 2.44952 0.492391 −3.58282 −1.00000 −0.948212 −0.115054 −1.03864
1.3 −1.27656 −2.57603 −0.370384 −4.09144 3.28847 −1.00000 3.02595 3.63595 5.22298
1.4 −0.120360 2.76784 −1.98551 −2.80853 −0.333137 −1.00000 0.479696 4.66094 0.338034
1.5 0.823556 −0.960649 −1.32176 2.98565 −0.791148 −1.00000 −2.73565 −2.07715 2.45885
1.6 1.38595 −0.122479 −0.0791355 −0.133004 −0.169750 −1.00000 −2.88158 −2.98500 −0.184338
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$1$$
$$11$$ $$1$$

## 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 847.2.a.m 6
3.b odd 2 1 7623.2.a.cs 6
7.b odd 2 1 5929.2.a.bj 6
11.b odd 2 1 847.2.a.n yes 6
11.c even 5 4 847.2.f.z 24
11.d odd 10 4 847.2.f.y 24
33.d even 2 1 7623.2.a.cp 6
77.b even 2 1 5929.2.a.bm 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.m 6 1.a even 1 1 trivial
847.2.a.n yes 6 11.b odd 2 1
847.2.f.y 24 11.d odd 10 4
847.2.f.z 24 11.c even 5 4
5929.2.a.bj 6 7.b odd 2 1
5929.2.a.bm 6 77.b even 2 1
7623.2.a.cp 6 33.d even 2 1
7623.2.a.cs 6 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(847))$$:

 $$T_{2}^{6} + 4T_{2}^{5} - 12T_{2}^{3} - 4T_{2}^{2} + 8T_{2} + 1$$ T2^6 + 4*T2^5 - 12*T2^3 - 4*T2^2 + 8*T2 + 1 $$T_{3}^{6} + 2T_{3}^{5} - 11T_{3}^{4} - 20T_{3}^{3} + 25T_{3}^{2} + 36T_{3} + 4$$ T3^6 + 2*T3^5 - 11*T3^4 - 20*T3^3 + 25*T3^2 + 36*T3 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 4 T^{5} - 12 T^{3} - 4 T^{2} + \cdots + 1$$
$3$ $$T^{6} + 2 T^{5} - 11 T^{4} - 20 T^{3} + \cdots + 4$$
$5$ $$T^{6} + 4 T^{5} - 9 T^{4} - 36 T^{3} + \cdots + 1$$
$7$ $$(T + 1)^{6}$$
$11$ $$T^{6}$$
$13$ $$T^{6} + 4 T^{5} - 19 T^{4} - 72 T^{3} + \cdots + 16$$
$17$ $$T^{6} + 22 T^{5} + 182 T^{4} + \cdots - 2687$$
$19$ $$T^{6} + 6 T^{5} - 50 T^{4} - 142 T^{3} + \cdots + 592$$
$23$ $$T^{6} - 2 T^{5} - 85 T^{4} + \cdots + 8656$$
$29$ $$T^{6} + 12 T^{5} + 28 T^{4} + \cdots + 481$$
$31$ $$T^{6} + 2 T^{5} - 118 T^{4} + \cdots + 6736$$
$37$ $$T^{6} - 14 T^{5} - 35 T^{4} + \cdots + 2896$$
$41$ $$T^{6} + 26 T^{5} + 247 T^{4} + \cdots - 3803$$
$43$ $$T^{6} - 4 T^{5} - 121 T^{4} + \cdots - 2288$$
$47$ $$T^{6} + 16 T^{5} - 27 T^{4} + \cdots + 9088$$
$53$ $$T^{6} - 4 T^{5} - 80 T^{4} + \cdots + 2257$$
$59$ $$T^{6} + 4 T^{5} - 287 T^{4} + \cdots - 313856$$
$61$ $$T^{6} - 8 T^{5} - 323 T^{4} + \cdots - 971552$$
$67$ $$T^{6} - 6 T^{5} - 161 T^{4} + \cdots - 48896$$
$71$ $$T^{6} - 22 T^{5} + 107 T^{4} + \cdots - 2048$$
$73$ $$T^{6} + 14 T^{5} - 122 T^{4} + \cdots - 23024$$
$79$ $$T^{6} - 28 T^{5} + 127 T^{4} + \cdots - 183488$$
$83$ $$T^{6} + 22 T^{5} + 42 T^{4} + \cdots - 44012$$
$89$ $$T^{6} - 254 T^{4} + 1400 T^{3} + \cdots + 62137$$
$97$ $$T^{6} + 4 T^{5} - 153 T^{4} + \cdots - 62759$$