# Properties

 Label 847.2.f.s Level $847$ Weight $2$ Character orbit 847.f Analytic conductor $6.763$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 847.f (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.76332905120$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.159390625.1 Defining polynomial: $$x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25$$ x^8 - x^7 + 6*x^6 - 11*x^5 + 21*x^4 - 5*x^3 + 10*x^2 + 25*x + 25 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{5} + (\beta_{7} + \beta_{5}) q^{6} - \beta_{2} q^{7} + (2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_1 + 1) q^{8} + (\beta_{3} - \beta_{2} + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b3 + 1) * q^3 + (-b7 - b6 - b5 - b3 - b2 + b1) * q^4 + (-b7 - b6 - b3 - b2 + 1) * q^5 + (b7 + b5) * q^6 - b2 * q^7 + (2*b6 + b5 - b3 - b1 + 1) * q^8 + (b3 - b2 + 1) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{5} + (\beta_{7} + \beta_{5}) q^{6} - \beta_{2} q^{7} + (2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_1 + 1) q^{8} + (\beta_{3} - \beta_{2} + 1) q^{9} + (\beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \beta_{2} + 2) q^{10} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 1) q^{12} + (2 \beta_{7} + 2 \beta_{4} - 2 \beta_{2} - 3 \beta_1 + 2) q^{13} - \beta_{4} q^{14} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{15} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} + 1) q^{16} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - 2) q^{17} + ( - \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_1) q^{18} + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_1 - 1) q^{19} + (\beta_{7} + \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{20} + ( - \beta_{6} - \beta_{2}) q^{21} + ( - 2 \beta_{7} - \beta_{6} - \beta_{2} + 2 \beta_1 - 2) q^{23} + ( - 4 \beta_{3} - 3 \beta_{2} - \beta_1 + 3) q^{24} + (2 \beta_{6} + 3 \beta_{4} + \beta_{3} - 1) q^{25} + (\beta_{7} - 3 \beta_{6} + \beta_{5} - 3 \beta_{3} + \beta_{2} - \beta_1) q^{26} + ( - \beta_{6} + 3 \beta_{3} + 3 \beta_{2} + 1) q^{27} + (\beta_{7} + \beta_{3} + \beta_{2}) q^{28} + (3 \beta_{4} - 3 \beta_1) q^{29} + (\beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 3) q^{30} + (\beta_{7} + \beta_{4} - 3 \beta_{2} - 2 \beta_1 + 3) q^{31} + ( - 3 \beta_{6} - 3 \beta_{2} + 4) q^{32} + (2 \beta_{7} - 3 \beta_{6} - 3 \beta_{2} - 2 \beta_1 - 1) q^{34} + (\beta_{3} + \beta_1) q^{35} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 1) q^{36} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{37} + ( - \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - 7 \beta_{3} - 7 \beta_{2} + 3) q^{38} + (\beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{39} + ( - 3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 4 \beta_{2}) q^{40} + (2 \beta_{6} - 2 \beta_{5} + 3 \beta_{3} + 2 \beta_1 - 3) q^{41} + ( - \beta_{7} - \beta_{4}) q^{42} + ( - 6 \beta_{6} - 6 \beta_{2} + 1) q^{43} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 + 2) q^{45} + ( - 3 \beta_{7} - 3 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 4) q^{46} + (\beta_{6} - 4 \beta_{5} - \beta_{4} - 2 \beta_{3} + 4 \beta_1 + 2) q^{47} + (3 \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1) q^{48} + (\beta_{6} + \beta_{3} + \beta_{2} - 1) q^{49} + ( - 2 \beta_{7} - 6 \beta_{6} - \beta_{5} - 9 \beta_{3} - 9 \beta_{2} + 6) q^{50} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - 4 \beta_{2} - \beta_1) q^{51} + ( - 6 \beta_{6} - \beta_{5} - 3 \beta_{3} + \beta_1 + 3) q^{52} + ( - 3 \beta_{7} - 3 \beta_{4} + 4 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{53} + ( - 4 \beta_{7} - 3 \beta_{5} + 3 \beta_{4} + 4 \beta_1) q^{54} + ( - \beta_{7} - \beta_{6} - \beta_{2} + \beta_1 - 2) q^{56} + (2 \beta_{7} + 2 \beta_{4} - \beta_{2} + \beta_1 + 1) q^{57} + ( - 3 \beta_{6} + 3 \beta_{5} - 6 \beta_{3} - 3 \beta_1 + 6) q^{58} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 6 \beta_{2} + \beta_1) q^{59} + (4 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 6 \beta_{3} + 6 \beta_{2} - 2) q^{60} + ( - 5 \beta_{7} + 7 \beta_{6} - \beta_{5} + 7 \beta_{3} + 7 \beta_{2} - 7) q^{61} + (\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{62} + (2 \beta_{6} + \beta_{3} - 1) q^{63} + (\beta_{7} + \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{64} + (2 \beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{2} - 2 \beta_1 - 2) q^{65} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{2} - \beta_1 - 2) q^{67} + (\beta_{7} + \beta_{4} - 2 \beta_{2} - 3 \beta_1 + 2) q^{68} + ( - \beta_{6} + 2 \beta_{4} + 3 \beta_{3} - 3) q^{69} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{70} + ( - 2 \beta_{7} - 4 \beta_{6} - 3 \beta_{5} + \beta_{3} + \beta_{2} + 4) q^{71} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 3) q^{72} + (2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - 5 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{73} + (3 \beta_{6} + \beta_{4} + \beta_{3} - 1) q^{74} + (3 \beta_{7} + 3 \beta_{4} - \beta_{2} + 1) q^{75} + (4 \beta_{7} + 3 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} + 3 \beta_{2} - 4 \beta_1 + 4) q^{76} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 - 3) q^{78} + ( - 2 \beta_{7} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{79} + ( - \beta_{6} + 4 \beta_{5} - \beta_{4} - 5 \beta_{3} - 4 \beta_1 + 5) q^{80} + (6 \beta_{6} + 6 \beta_{3} + 4 \beta_{2}) q^{81} + ( - \beta_{7} + 2 \beta_{6} - 5 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{82} + ( - 3 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} + 4 \beta_{3} + 4 \beta_{2} + 3) q^{83} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2}) q^{84} + (3 \beta_{6} - \beta_{5} - \beta_{4} + \beta_1) q^{85} + ( - 6 \beta_{7} - 6 \beta_{4} + \beta_1) q^{86} + ( - 3 \beta_{5} + 3 \beta_{4}) q^{87} + (5 \beta_{7} - 3 \beta_{6} - \beta_{5} + \beta_{4} - 3 \beta_{2} - 5 \beta_1 - 1) q^{89} + ( - 2 \beta_{7} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 4 \beta_1 + 1) q^{90} + (2 \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 2) q^{91} + (5 \beta_{7} + 7 \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + 7 \beta_{3} + 6 \beta_{2} - 2 \beta_1) q^{92} + ( - 3 \beta_{6} - \beta_{5} - 3 \beta_{3} - 3 \beta_{2} + 3) q^{93} + (4 \beta_{7} + 6 \beta_{6} - 2 \beta_{5} - \beta_{3} - \beta_{2} - 6) q^{94} + (3 \beta_{7} + 5 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} + 6 \beta_{2}) q^{95} + ( - 3 \beta_{6} - \beta_{3} + 1) q^{96} + (5 \beta_{7} + 5 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 2) q^{97} + ( - \beta_{5} + \beta_{4}) q^{98}+O(q^{100})$$ q + b1 * q^2 + (-b3 + 1) * q^3 + (-b7 - b6 - b5 - b3 - b2 + b1) * q^4 + (-b7 - b6 - b3 - b2 + 1) * q^5 + (b7 + b5) * q^6 - b2 * q^7 + (2*b6 + b5 - b3 - b1 + 1) * q^8 + (b3 - b2 + 1) * q^9 + (b6 + 2*b5 - 2*b4 + b2 + 2) * q^10 + (-b7 - b6 - b5 + b4 - b2 + b1 - 1) * q^12 + (2*b7 + 2*b4 - 2*b2 - 3*b1 + 2) * q^13 - b4 * q^14 + (-b7 - b6 - b5 + b4 - b3 - b2) * q^15 + (b7 - b6 + 2*b5 - b3 - b2 + 1) * q^16 + (b7 + 2*b6 + b5 - b3 - b2 - 2) * q^17 + (-b7 - b5 - b4 + 2*b1) * q^18 + (2*b6 - b5 + 2*b4 + b3 + b1 - 1) * q^19 + (b7 + b4 + 4*b3 + 2*b2 + 2*b1 - 2) * q^20 + (-b6 - b2) * q^21 + (-2*b7 - b6 - b2 + 2*b1 - 2) * q^23 + (-4*b3 - 3*b2 - b1 + 3) * q^24 + (2*b6 + 3*b4 + b3 - 1) * q^25 + (b7 - 3*b6 + b5 - 3*b3 + b2 - b1) * q^26 + (-b6 + 3*b3 + 3*b2 + 1) * q^27 + (b7 + b3 + b2) * q^28 + (3*b4 - 3*b1) * q^29 + (b6 + 2*b5 - 2*b4 - 3*b3 - 2*b1 + 3) * q^30 + (b7 + b4 - 3*b2 - 2*b1 + 3) * q^31 + (-3*b6 - 3*b2 + 4) * q^32 + (2*b7 - 3*b6 - 3*b2 - 2*b1 - 1) * q^34 + (b3 + b1) * q^35 + (-b6 - b5 - b4 - b3 + b1 + 1) * q^36 + (-b7 - b6 - b5 - b4 - b3 + 2*b2 + 2*b1) * q^37 + (-b7 - 3*b6 - 2*b5 - 7*b3 - 7*b2 + 3) * q^38 + (b7 - 2*b6 - b5 - 2*b3 - 2*b2 + 2) * q^39 + (-3*b7 - 3*b6 - 3*b5 + 3*b4 - 3*b3 - 4*b2) * q^40 + (2*b6 - 2*b5 + 3*b3 + 2*b1 - 3) * q^41 + (-b7 - b4) * q^42 + (-6*b6 - 6*b2 + 1) * q^43 + (-b7 - b6 + b5 - b4 - b2 + b1 + 2) * q^45 + (-3*b7 - 3*b4 - 2*b3 - 4*b2 + 4) * q^46 + (b6 - 4*b5 - b4 - 2*b3 + 4*b1 + 2) * q^47 + (3*b7 - b6 + 3*b5 - b4 - b3 - b2 - 2*b1) * q^48 + (b6 + b3 + b2 - 1) * q^49 + (-2*b7 - 6*b6 - b5 - 9*b3 - 9*b2 + 6) * q^50 + (2*b7 - b6 + 2*b5 - b4 - b3 - 4*b2 - b1) * q^51 + (-6*b6 - b5 - 3*b3 + b1 + 3) * q^52 + (-3*b7 - 3*b4 + 4*b3 + b2 - b1 - 1) * q^53 + (-4*b7 - 3*b5 + 3*b4 + 4*b1) * q^54 + (-b7 - b6 - b2 + b1 - 2) * q^56 + (2*b7 + 2*b4 - b2 + b1 + 1) * q^57 + (-3*b6 + 3*b5 - 6*b3 - 3*b1 + 6) * q^58 + (-2*b7 - 2*b6 - 2*b5 + b4 - 2*b3 + 6*b2 + b1) * q^59 + (4*b7 + 2*b6 + 3*b5 + 6*b3 + 6*b2 - 2) * q^60 + (-5*b7 + 7*b6 - b5 + 7*b3 + 7*b2 - 7) * q^61 + (b7 - b6 + b5 - 2*b4 - b3 + 2*b2 + b1) * q^62 + (2*b6 + b3 - 1) * q^63 + (b7 + b4 - 2*b3 + 2*b1) * q^64 + (2*b7 + b6 - 2*b5 + 2*b4 + b2 - 2*b1 - 2) * q^65 + (b7 - 2*b6 - b5 + b4 - 2*b2 - b1 - 2) * q^67 + (b7 + b4 - 2*b2 - 3*b1 + 2) * q^68 + (-b6 + 2*b4 + 3*b3 - 3) * q^69 + (-2*b7 - b6 - 2*b5 - b3 - 3*b2 + 2*b1) * q^70 + (-2*b7 - 4*b6 - 3*b5 + b3 + b2 + 4) * q^71 + (-b7 + 3*b6 + 2*b5 + 2*b3 + 2*b2 - 3) * q^72 + (2*b7 - 3*b6 + 2*b5 - 5*b4 - 3*b3 - 4*b2 + 3*b1) * q^73 + (3*b6 + b4 + b3 - 1) * q^74 + (3*b7 + 3*b4 - b2 + 1) * q^75 + (4*b7 + 3*b6 + 4*b5 - 4*b4 + 3*b2 - 4*b1 + 4) * q^76 + (b7 + b6 + b5 - b4 + b2 - b1 - 3) * q^78 + (-2*b7 - 2*b4 + 2*b3 + 2*b2 + 3*b1 - 2) * q^79 + (-b6 + 4*b5 - b4 - 5*b3 - 4*b1 + 5) * q^80 + (6*b6 + 6*b3 + 4*b2) * q^81 + (-b7 + 2*b6 - 5*b5 - 2*b3 - 2*b2 - 2) * q^82 + (-3*b7 - 3*b6 - 5*b5 + 4*b3 + 4*b2 + 3) * q^83 + (b7 + b6 + b5 - b4 + b3 + 2*b2) * q^84 + (3*b6 - b5 - b4 + b1) * q^85 + (-6*b7 - 6*b4 + b1) * q^86 + (-3*b5 + 3*b4) * q^87 + (5*b7 - 3*b6 - b5 + b4 - 3*b2 - 5*b1 - 1) * q^89 + (-2*b7 - 2*b4 + 2*b3 - b2 + 4*b1 + 1) * q^90 + (2*b6 + 2*b5 + b4 + 2*b3 - 2*b1 - 2) * q^91 + (5*b7 + 7*b6 + 5*b5 - 3*b4 + 7*b3 + 6*b2 - 2*b1) * q^92 + (-3*b6 - b5 - 3*b3 - 3*b2 + 3) * q^93 + (4*b7 + 6*b6 - 2*b5 - b3 - b2 - 6) * q^94 + (3*b7 + 5*b6 + 3*b5 - 3*b4 + 5*b3 + 6*b2) * q^95 + (-3*b6 - b3 + 1) * q^96 + (5*b7 + 5*b4 + 6*b3 + 2*b2 - 5*b1 - 2) * q^97 + (-b5 + b4) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{2} + 6 q^{3} - 7 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10})$$ 8 * q + q^2 + 6 * q^3 - 7 * q^4 + 3 * q^5 + 2 * q^6 - 2 * q^7 + 12 * q^8 + 8 * q^9 $$8 q + q^{2} + 6 q^{3} - 7 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{7} + 12 q^{8} + 8 q^{9} + 28 q^{10} - 14 q^{12} + 5 q^{13} + q^{14} - 9 q^{15} + 7 q^{16} - 14 q^{17} + q^{18} - 6 q^{19} - 4 q^{20} - 4 q^{21} - 16 q^{23} + 9 q^{24} - 5 q^{25} - 9 q^{26} + 18 q^{27} + 3 q^{28} - 6 q^{29} + 26 q^{30} + 14 q^{31} + 20 q^{32} - 24 q^{34} + 3 q^{35} + 3 q^{36} + q^{37} - 15 q^{38} - 29 q^{40} - 18 q^{41} + 2 q^{42} - 16 q^{43} + 18 q^{45} + 26 q^{46} + 7 q^{47} - q^{48} - 2 q^{49} - q^{50} - 8 q^{51} + 4 q^{52} + 7 q^{53} - 4 q^{54} - 18 q^{56} + 3 q^{57} + 36 q^{58} + 17 q^{60} - 12 q^{61} + 5 q^{62} - 2 q^{63} - 4 q^{64} - 24 q^{65} - 30 q^{67} + 7 q^{68} - 22 q^{69} - 12 q^{70} + 21 q^{71} - 3 q^{72} - 8 q^{73} - q^{74} + 52 q^{76} - 18 q^{78} - q^{79} + 37 q^{80} + 32 q^{81} - 34 q^{82} + 22 q^{83} + 11 q^{84} + 5 q^{85} + 13 q^{86} - 12 q^{87} - 34 q^{89} + 18 q^{90} - 5 q^{91} + 51 q^{92} + 3 q^{93} - 50 q^{94} + 41 q^{95} - 15 q^{97} - 4 q^{98}+O(q^{100})$$ 8 * q + q^2 + 6 * q^3 - 7 * q^4 + 3 * q^5 + 2 * q^6 - 2 * q^7 + 12 * q^8 + 8 * q^9 + 28 * q^10 - 14 * q^12 + 5 * q^13 + q^14 - 9 * q^15 + 7 * q^16 - 14 * q^17 + q^18 - 6 * q^19 - 4 * q^20 - 4 * q^21 - 16 * q^23 + 9 * q^24 - 5 * q^25 - 9 * q^26 + 18 * q^27 + 3 * q^28 - 6 * q^29 + 26 * q^30 + 14 * q^31 + 20 * q^32 - 24 * q^34 + 3 * q^35 + 3 * q^36 + q^37 - 15 * q^38 - 29 * q^40 - 18 * q^41 + 2 * q^42 - 16 * q^43 + 18 * q^45 + 26 * q^46 + 7 * q^47 - q^48 - 2 * q^49 - q^50 - 8 * q^51 + 4 * q^52 + 7 * q^53 - 4 * q^54 - 18 * q^56 + 3 * q^57 + 36 * q^58 + 17 * q^60 - 12 * q^61 + 5 * q^62 - 2 * q^63 - 4 * q^64 - 24 * q^65 - 30 * q^67 + 7 * q^68 - 22 * q^69 - 12 * q^70 + 21 * q^71 - 3 * q^72 - 8 * q^73 - q^74 + 52 * q^76 - 18 * q^78 - q^79 + 37 * q^80 + 32 * q^81 - 34 * q^82 + 22 * q^83 + 11 * q^84 + 5 * q^85 + 13 * q^86 - 12 * q^87 - 34 * q^89 + 18 * q^90 - 5 * q^91 + 51 * q^92 + 3 * q^93 - 50 * q^94 + 41 * q^95 - 15 * q^97 - 4 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 555\nu^{7} - 2159\nu^{6} + 7489\nu^{5} - 18164\nu^{4} + 40069\nu^{3} - 84434\nu^{2} + 43855\nu + 375 ) / 94655$$ (555*v^7 - 2159*v^6 + 7489*v^5 - 18164*v^4 + 40069*v^3 - 84434*v^2 + 43855*v + 375) / 94655 $$\beta_{3}$$ $$=$$ $$( -970\nu^{7} - 1002\nu^{6} - 6608\nu^{5} + 9063\nu^{4} - 14943\nu^{3} + 27673\nu^{2} - 68120\nu + 35160 ) / 94655$$ (-970*v^7 - 1002*v^6 - 6608*v^5 + 9063*v^4 - 14943*v^3 + 27673*v^2 - 68120*v + 35160) / 94655 $$\beta_{4}$$ $$=$$ $$( -1604\nu^{7} + 4159\nu^{6} - 12059\nu^{5} + 28414\nu^{4} - 81659\nu^{3} + 38305\nu^{2} - 13500\nu - 13875 ) / 94655$$ (-1604*v^7 + 4159*v^6 - 12059*v^5 + 28414*v^4 - 81659*v^3 + 38305*v^2 - 13500*v - 13875) / 94655 $$\beta_{5}$$ $$=$$ $$( -2052\nu^{7} + 2252\nu^{6} - 19912\nu^{5} + 21007\nu^{4} - 82042\nu^{3} + 35785\nu^{2} - 19395\nu - 90925 ) / 94655$$ (-2052*v^7 + 2252*v^6 - 19912*v^5 + 21007*v^4 - 82042*v^3 + 35785*v^2 - 19395*v - 90925) / 94655 $$\beta_{6}$$ $$=$$ $$( -2667\nu^{7} + 6691\nu^{6} - 17466\nu^{5} + 50856\nu^{4} - 82441\nu^{3} + 72554\nu^{2} - 4035\nu - 12035 ) / 94655$$ (-2667*v^7 + 6691*v^6 - 17466*v^5 + 50856*v^4 - 82441*v^3 + 72554*v^2 - 4035*v - 12035) / 94655 $$\beta_{7}$$ $$=$$ $$( 4024\nu^{7} - 1464\nu^{6} + 21519\nu^{5} - 26434\nu^{4} + 59219\nu^{3} + 22635\nu^{2} + 54640\nu + 66675 ) / 94655$$ (4024*v^7 - 1464*v^6 + 21519*v^5 - 26434*v^4 + 59219*v^3 + 22635*v^2 + 54640*v + 66675) / 94655
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 3\beta_{2} + \beta_1$$ -b7 - b6 - b5 - b3 - 3*b2 + b1 $$\nu^{3}$$ $$=$$ $$2\beta_{6} + \beta_{5} - 4\beta_{4} - \beta_{3} - \beta _1 + 1$$ 2*b6 + b5 - 4*b4 - b3 - b1 + 1 $$\nu^{4}$$ $$=$$ $$7\beta_{7} + 7\beta_{6} + 2\beta_{5} + 13\beta_{3} + 13\beta_{2} - 7$$ 7*b7 + 7*b6 + 2*b5 + 13*b3 + 13*b2 - 7 $$\nu^{5}$$ $$=$$ $$-8\beta_{7} - 11\beta_{6} - 20\beta_{5} + 20\beta_{4} - 11\beta_{2} + 8\beta _1 - 12$$ -8*b7 - 11*b6 - 20*b5 + 20*b4 - 11*b2 + 8*b1 - 12 $$\nu^{6}$$ $$=$$ $$-19\beta_{7} - 19\beta_{4} - 68\beta_{3} - 36\beta_{2} - 24\beta _1 + 36$$ -19*b7 - 19*b4 - 68*b3 - 36*b2 - 24*b1 + 36 $$\nu^{7}$$ $$=$$ $$111\beta_{7} + 81\beta_{6} + 111\beta_{5} - 55\beta_{4} + 81\beta_{3} + 148\beta_{2} - 56\beta_1$$ 111*b7 + 81*b6 + 111*b5 - 55*b4 + 81*b3 + 148*b2 - 56*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$365$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
148.1
 −0.762262 − 2.34600i 0.453245 + 1.39494i −0.628998 − 0.456994i 1.43801 + 1.04478i −0.762262 + 2.34600i 0.453245 − 1.39494i −0.628998 + 0.456994i 1.43801 − 1.04478i
−0.762262 2.34600i 1.30902 + 0.951057i −3.30464 + 2.40097i −1.07128 + 3.29706i 1.23337 3.79591i −0.809017 + 0.587785i 4.16042 + 3.02272i −0.118034 0.363271i 8.55150
148.2 0.453245 + 1.39494i 1.30902 + 0.951057i −0.122406 + 0.0889332i 0.144228 0.443888i −0.733366 + 2.25707i −0.809017 + 0.587785i 2.19369 + 1.59381i −0.118034 0.363271i 0.684570
323.1 −0.628998 0.456994i 0.190983 0.587785i −0.431239 1.32722i 0.180019 0.130791i −0.388742 + 0.282438i 0.309017 + 0.951057i −0.815793 + 2.51075i 2.11803 + 1.53884i −0.173002
323.2 1.43801 + 1.04478i 0.190983 0.587785i 0.358290 + 1.10270i 2.24703 1.63256i 0.888742 0.645709i 0.309017 + 0.951057i 0.461691 1.42094i 2.11803 + 1.53884i 4.93693
372.1 −0.762262 + 2.34600i 1.30902 0.951057i −3.30464 2.40097i −1.07128 3.29706i 1.23337 + 3.79591i −0.809017 0.587785i 4.16042 3.02272i −0.118034 + 0.363271i 8.55150
372.2 0.453245 1.39494i 1.30902 0.951057i −0.122406 0.0889332i 0.144228 + 0.443888i −0.733366 2.25707i −0.809017 0.587785i 2.19369 1.59381i −0.118034 + 0.363271i 0.684570
729.1 −0.628998 + 0.456994i 0.190983 + 0.587785i −0.431239 + 1.32722i 0.180019 + 0.130791i −0.388742 0.282438i 0.309017 0.951057i −0.815793 2.51075i 2.11803 1.53884i −0.173002
729.2 1.43801 1.04478i 0.190983 + 0.587785i 0.358290 1.10270i 2.24703 + 1.63256i 0.888742 + 0.645709i 0.309017 0.951057i 0.461691 + 1.42094i 2.11803 1.53884i 4.93693
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 729.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.f.s 8
11.b odd 2 1 847.2.f.p 8
11.c even 5 1 847.2.a.k 4
11.c even 5 2 847.2.f.q 8
11.c even 5 1 inner 847.2.f.s 8
11.d odd 10 2 77.2.f.a 8
11.d odd 10 1 847.2.a.l 4
11.d odd 10 1 847.2.f.p 8
33.f even 10 2 693.2.m.g 8
33.f even 10 1 7623.2.a.ch 4
33.h odd 10 1 7623.2.a.co 4
77.j odd 10 1 5929.2.a.bb 4
77.l even 10 2 539.2.f.d 8
77.l even 10 1 5929.2.a.bi 4
77.n even 30 4 539.2.q.b 16
77.o odd 30 4 539.2.q.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.a 8 11.d odd 10 2
539.2.f.d 8 77.l even 10 2
539.2.q.b 16 77.n even 30 4
539.2.q.c 16 77.o odd 30 4
693.2.m.g 8 33.f even 10 2
847.2.a.k 4 11.c even 5 1
847.2.a.l 4 11.d odd 10 1
847.2.f.p 8 11.b odd 2 1
847.2.f.p 8 11.d odd 10 1
847.2.f.q 8 11.c even 5 2
847.2.f.s 8 1.a even 1 1 trivial
847.2.f.s 8 11.c even 5 1 inner
5929.2.a.bb 4 77.j odd 10 1
5929.2.a.bi 4 77.l even 10 1
7623.2.a.ch 4 33.f even 10 1
7623.2.a.co 4 33.h odd 10 1

## Hecke kernels

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

 $$T_{2}^{8} - T_{2}^{7} + 6T_{2}^{6} - 11T_{2}^{5} + 21T_{2}^{4} - 5T_{2}^{3} + 10T_{2}^{2} + 25T_{2} + 25$$ T2^8 - T2^7 + 6*T2^6 - 11*T2^5 + 21*T2^4 - 5*T2^3 + 10*T2^2 + 25*T2 + 25 $$T_{3}^{4} - 3T_{3}^{3} + 4T_{3}^{2} - 2T_{3} + 1$$ T3^4 - 3*T3^3 + 4*T3^2 - 2*T3 + 1 $$T_{13}^{8} - 5T_{13}^{7} - 4T_{13}^{6} + 105T_{13}^{5} + 611T_{13}^{4} - 735T_{13}^{3} + 2456T_{13}^{2} - 1015T_{13} + 841$$ T13^8 - 5*T13^7 - 4*T13^6 + 105*T13^5 + 611*T13^4 - 735*T13^3 + 2456*T13^2 - 1015*T13 + 841

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{7} + 6 T^{6} - 11 T^{5} + \cdots + 25$$
$3$ $$(T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1)^{2}$$
$5$ $$T^{8} - 3 T^{7} + 12 T^{6} - 45 T^{5} + \cdots + 1$$
$7$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$11$ $$T^{8}$$
$13$ $$T^{8} - 5 T^{7} - 4 T^{6} + 105 T^{5} + \cdots + 841$$
$17$ $$T^{8} + 14 T^{7} + 113 T^{6} + \cdots + 5041$$
$19$ $$T^{8} + 6 T^{7} + 11 T^{6} + \cdots + 21025$$
$23$ $$(T^{4} + 8 T^{3} - 9 T^{2} - 150 T - 205)^{2}$$
$29$ $$T^{8} + 6 T^{7} + 81 T^{6} + \cdots + 164025$$
$31$ $$T^{8} - 14 T^{7} + 91 T^{6} - 279 T^{5} + \cdots + 25$$
$37$ $$T^{8} - T^{7} + 26 T^{6} - 121 T^{5} + \cdots + 25$$
$41$ $$T^{8} + 18 T^{7} + 227 T^{6} + \cdots + 6241$$
$43$ $$(T^{2} + 4 T - 41)^{4}$$
$47$ $$T^{8} - 7 T^{7} + 179 T^{6} + \cdots + 93025$$
$53$ $$T^{8} - 7 T^{7} + 238 T^{6} + \cdots + 755161$$
$59$ $$T^{8} + 109 T^{6} - 675 T^{5} + \cdots + 1413721$$
$61$ $$T^{8} + 12 T^{7} + 269 T^{6} + \cdots + 990025$$
$67$ $$(T^{4} + 15 T^{3} + 67 T^{2} + 45 T - 199)^{2}$$
$71$ $$T^{8} - 21 T^{7} + 293 T^{6} + \cdots + 982081$$
$73$ $$T^{8} + 8 T^{7} + 69 T^{6} + \cdots + 24750625$$
$79$ $$T^{8} + T^{7} - 2 T^{6} - 5 T^{5} + \cdots + 73441$$
$83$ $$T^{8} - 22 T^{7} + 479 T^{6} + \cdots + 5040025$$
$89$ $$(T^{4} + 17 T^{3} - 44 T^{2} - 1120 T - 755)^{2}$$
$97$ $$T^{8} + 15 T^{7} + 230 T^{6} + \cdots + 17850625$$