# Properties

 Label 847.2.f.x Level $847$ Weight $2$ Character orbit 847.f Analytic conductor $6.763$ Analytic rank $0$ Dimension $16$ 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: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} - 1175 x^{7} + 2135 x^{6} - 2300 x^{5} + 1850 x^{4} - 925 x^{3} + 700 x^{2} + \cdots + 25$$ x^16 - 3*x^15 + 14*x^14 - 32*x^13 + 86*x^12 - 145*x^11 + 245*x^10 - 245*x^9 + 640*x^8 - 1175*x^7 + 2135*x^6 - 2300*x^5 + 1850*x^4 - 925*x^3 + 700*x^2 - 200*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} + \beta_{5}) q^{2} + (\beta_{13} - \beta_{11} + \beta_{9}) q^{3} + (\beta_{11} + \beta_{7} + \beta_{2}) q^{4} + (\beta_{13} - \beta_{7} + \beta_{6} - \beta_{3} + \beta_1) q^{5} + (\beta_{13} - \beta_{11} + \beta_{10} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{6} + ( - \beta_{10} + \beta_{8} - \beta_{7} + 1) q^{7} + ( - \beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{2} + \beta_1) q^{8} + (\beta_{14} - \beta_{12} - \beta_{9} + \beta_{8} - \beta_{3}) q^{9}+O(q^{10})$$ q + (b6 + b5) * q^2 + (b13 - b11 + b9) * q^3 + (b11 + b7 + b2) * q^4 + (b13 - b7 + b6 - b3 + b1) * q^5 + (b13 - b11 + b10 + b6 - b5 - b4 - b3 + 2*b1 - 1) * q^6 + (-b10 + b8 - b7 + 1) * q^7 + (-b15 + b13 + b12 - b10 + b8 - b6 - b5 - b2 + b1) * q^8 + (b14 - b12 - b9 + b8 - b3) * q^9 $$q + (\beta_{6} + \beta_{5}) q^{2} + (\beta_{13} - \beta_{11} + \beta_{9}) q^{3} + (\beta_{11} + \beta_{7} + \beta_{2}) q^{4} + (\beta_{13} - \beta_{7} + \beta_{6} - \beta_{3} + \beta_1) q^{5} + (\beta_{13} - \beta_{11} + \beta_{10} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{6} + ( - \beta_{10} + \beta_{8} - \beta_{7} + 1) q^{7} + ( - \beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{2} + \beta_1) q^{8} + (\beta_{14} - \beta_{12} - \beta_{9} + \beta_{8} - \beta_{3}) q^{9} + ( - 2 \beta_{8} + 2 \beta_{7} - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{10} + (\beta_{12} - \beta_{11} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} - \beta_{2} - 1) q^{12} + ( - \beta_{15} + \beta_{13} + \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{4} - \beta_{3} - \beta_1) q^{13} + \beta_{6} q^{14} + ( - \beta_{15} + \beta_{14} + 2 \beta_{12} - 2 \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{5} + \cdots - 4) q^{15}+ \cdots - \beta_1 q^{98}+O(q^{100})$$ q + (b6 + b5) * q^2 + (b13 - b11 + b9) * q^3 + (b11 + b7 + b2) * q^4 + (b13 - b7 + b6 - b3 + b1) * q^5 + (b13 - b11 + b10 + b6 - b5 - b4 - b3 + 2*b1 - 1) * q^6 + (-b10 + b8 - b7 + 1) * q^7 + (-b15 + b13 + b12 - b10 + b8 - b6 - b5 - b2 + b1) * q^8 + (b14 - b12 - b9 + b8 - b3) * q^9 + (-2*b8 + 2*b7 - b3 + b2 - 2*b1 - 1) * q^10 + (b12 - b11 - b8 + b7 + b6 - b4 - b2 - 1) * q^12 + (-b15 + b13 + b10 + b9 - b8 + 2*b7 + b4 - b3 - b1) * q^13 + b6 * q^14 + (-b15 + b14 + 2*b12 - 2*b9 - b8 + 2*b7 - b5 + b4 - b3 - 2*b2 - 4) * q^15 + (-b15 - b14 - b11 - 2*b10 - b7 + b5 + b4 - b1 + 1) * q^16 + (b15 + b14 - b13 - 2*b7 + b6 + b5 + b4 + b3 + 1) * q^17 + (-b15 - b14 - b12 + b11 - b10 + b9 + b7 - 2*b5 - b3 + 2*b2 + 1) * q^18 + (b15 + b12 - b10 - b8 + b6 - b2 - 2) * q^19 + (-b14 + b12 + b10 + 2*b9 + 3*b8 + b7 - b6 - b5 - b4 + b3 + b1) * q^20 + (b3 - b2 - 1) * q^21 + (b14 + b13 + b12 - b11 - b8 + b7 + 2*b6 - 2*b4 - 2*b3 - b2 + 2*b1) * q^23 + (b15 - b14 - b13 - b12 + 5*b10 - b8 + 4*b7 + b4 + 2*b3 - b1) * q^24 + (-b15 - b13 + b11 - 5*b10 - b9 + 2*b8 + 3*b5 - 3*b1 + 2) * q^25 + (-b12 - b11 - b10 + b9 + b8 + b7 + b5 + 2*b4 - 1) * q^26 + (-b15 - b14 - b13 + 2*b11 - 3*b10 + b9 - 2*b7 + 2*b5 + 2*b4 + b3 + b2 - 2*b1 + 3) * q^27 + (b10 + b9 + b7 + b2) * q^28 + (-b15 + 2*b12 + b11 - 4*b10 - 2*b9 + 3*b8 - b7 - b4 - b3 - b2) * q^29 + (-b15 - 2*b13 - b12 + b11 - 7*b10 - b9 + 2*b8 - 2*b6 + 2*b5 + b2 - 2*b1 + 3) * q^30 + (2*b15 - 2*b13 + b10 - b9 + b8 - b7 + b6 + b5 + 3*b4 + 2*b3 - 3*b1) * q^31 + (b14 + b13 + b12 - b11 + 2*b6 - 2*b4 - b3 + 2*b1 - 3) * q^32 + (b14 + b13 - b12 + b11 - 2*b8 + 2*b7 + b3 + 2*b2) * q^34 + (-b14 - b8 + b4 + b3 - b1) * q^35 + (2*b15 - b12 - b11 + 2*b10 + b9 + 4*b8 - b6 - 2*b5 + b2 + 2*b1 + 5) * q^36 + (b15 - 2*b14 - b12 + 2*b10 + b9 - b8 + b7 - b5 - b4 + b3 + b2) * q^37 + (2*b15 + 2*b14 - b13 + 2*b10 + b7 - b6 + b3 - b1) * q^38 + (-2*b15 - 2*b14 - b13 + b11 - b10 - 2*b9 + 4*b7 - 3*b6 + b5 + b4 + b3 - 2*b2 - 4*b1 - 3) * q^39 + (2*b15 - 3*b11 + b10 + b8 - 2*b7 + 3*b5 + 3*b4 + 2*b3 - 3*b2 - 1) * q^40 + (-b13 - 2*b12 - b11 + b9 + 2*b2 + 2) * q^41 + (-b14 - b12 + b10 - b8 + b7 - b6 - b5 + b4 + b3 - b1) * q^42 + (-b14 - b13 - b12 + b11 - b8 + b7 - 2*b6 + 2*b4 - b3 + 2*b2 - 2*b1 + 2) * q^43 + (-3*b14 - 3*b13 - 4*b12 + 4*b11 - 5*b8 + 5*b7 - b6 + b4 + 2*b3 + 4*b2 - 2*b1 + 6) * q^45 + (b15 + 2*b14 - b13 + 2*b12 + 3*b10 - b9 - b8 + 2*b7 + 3*b6 + 3*b5 - b3) * q^46 + (-b15 + 2*b12 + 2*b11 - 4*b10 - 2*b9 - 2*b6 - b5 - 2*b2 + b1 - 2) * q^47 + (b15 - b14 - 2*b12 - 2*b11 + 2*b9 + b8 + 2*b7 + b5 + b4 + b3 - 2) * q^48 - b7 * q^49 + (-b15 - b14 - b13 + 4*b11 + 4*b10 + 4*b7 + b6 - 2*b5 - 2*b4 + b3 + 3*b1 - 5) * q^50 + (2*b15 + b14 - 2*b11 + 2*b10 - 3*b7 - 2*b5 - b4 + 2*b3 - 2*b2 + 1) * q^51 + (b15 + 2*b13 - b12 - 3*b10 - b8 + b6 - 2*b5 + b2 + 2*b1) * q^52 + (-2*b15 + 2*b13 + 3*b12 + 5*b8 + 2*b7 - b6 - b5 - 2*b3) * q^53 + (-b12 + b11 - 2*b8 + 2*b7 - b6 + b4 + 3*b3 - b2 - b1 - 4) * q^54 + (b14 + b13 + b12 - b11 + b8 - b7 - b6 + b4 - b2) * q^56 + (3*b15 + 3*b14 - 3*b13 - b12 + 3*b10 - 4*b9 - b8 - 2*b6 - 2*b5 - b4 + b1) * q^57 + (-b15 - 2*b13 + 2*b12 + 2*b11 - 2*b10 - 2*b9 + 2*b8 + 2*b5 - 2*b2 - 2*b1) * q^58 + (-b15 + b14 + b12 + b11 - b10 - b9 - b7 - 3*b4 - b3 + 1) * q^59 + (b15 + b14 + b13 - b11 + 4*b10 - b9 + b7 - 3*b5 - 3*b4 - b3 - b2 + 3*b1 - 4) * q^60 + (b15 + b14 - b11 + 2*b10 + b9 + 2*b7 - b6 - 3*b5 - 3*b4 + b2 + 2*b1) * q^61 + (b15 + b14 - 3*b12 + 2*b11 + 3*b9 + b8 + 5*b7 + b5 + b3 + 5*b2) * q^62 + (b15 - b12 + b11 + 2*b10 - b9 + b2 + 1) * q^63 + (-3*b15 + b14 + 3*b13 + b12 - 2*b10 - b9 + b7 - 2*b4 - 4*b3 + 2*b1) * q^64 + (-2*b8 + 2*b7 + 2*b6 - 2*b4 - b3 - 3*b2 + 4*b1 - 3) * q^65 + (-b14 - b13 - 2*b8 + 2*b7 - b6 + b4 + 2*b1 - 4) * q^67 + (-3*b15 - b14 + 3*b13 + b12 - 2*b10 + b9 + b7 - 5*b6 - 5*b5 - 4*b4 - 2*b3 + 4*b1) * q^68 + (b15 - 3*b12 - b11 + b10 + b9 - 4*b5 + 3*b2 + 4*b1 + 3) * q^69 + (-b14 - b12 + b9 + 2*b7 - 2*b4 + b2 - 1) * q^70 + (3*b15 + 3*b14 - 3*b13 - b11 + 3*b10 - 3*b9 - 3*b7 - b5 - b4 + 3*b3 - 3*b2 + b1 - 3) * q^71 + (3*b13 - b11 - 2*b10 + 2*b9 - 5*b7 + 2*b6 + 4*b5 + 4*b4 - 3*b3 + 2*b2 - 2*b1 + 4) * q^72 + (-2*b15 + b14 + 2*b11 + 4*b10 - 6*b8 + b7 - b4 - 2*b3 + 2*b2 + 1) * q^73 + (2*b15 + b12 - 2*b11 + 5*b10 + 2*b9 + b8 - b2) * q^74 + (-2*b14 - 2*b12 - b10 + b9 + b8 - b7 + 3*b6 + 3*b5 - 5*b4 + 2*b3 + 5*b1) * q^75 + (2*b12 - 2*b11 + 2*b8 - 2*b7 - 2*b6 + 2*b4 + 3*b3 - 3*b2 - 5) * q^76 + (-4*b14 - 4*b13 - 3*b12 + 3*b11 + b8 - b7 - b6 + b4 + 2*b3 - b2 - 2*b1 - 3) * q^78 + (-4*b15 + 4*b13 + b12 - 5*b10 - b9 + 2*b8 - b7 + 2*b6 + 2*b5 - b4 - 4*b3 + b1) * q^79 + (b15 - b13 - 2*b12 + b11 + 6*b10 - b9 - 10*b8 + 4*b6 - b5 + 2*b2 + b1 - 8) * q^80 + (b15 + 3*b14 - 3*b11 + 2*b10 - b8 + b7 - 4*b4 + b3 - 3*b2 - 4) * q^81 + (b13 - b11 + b10 + 2*b9 + 3*b6 - 3*b5 - 3*b4 - b3 + 2*b2 + 6*b1 + 1) * q^82 + (2*b15 + 2*b14 + 2*b13 - 2*b11 + b10 + b9 - 4*b7 + b6 + b5 + b4 - 2*b3 + b2 + 2) * q^83 + (-b11 - b10 + b8 - b5 - b4 - b2 - 1) * q^84 + (b13 - b12 - b11 + b10 + b9 - b8 + 3*b6 + 3*b5 + b2 - 3*b1) * q^85 + (b15 + b14 - b13 - b12 - 4*b10 + b9 + 2*b8 - 5*b7 + b6 + b5 - b4 + b1) * q^86 + (-2*b14 - 2*b13 - b12 + b11 - 4*b8 + 4*b7 + b6 - b4 + 3*b3 + b2 + 3*b1 + 2) * q^87 + (2*b14 + 2*b13 + 3*b12 - 3*b11 + b8 - b7 - b6 + b4 - 3*b2 - 4*b1 + 3) * q^89 + (-3*b14 - b12 + 2*b10 + 5*b9 + 2*b7 - 2*b6 - 2*b5 - 4*b4 + 3*b3 + 4*b1) * q^90 + (b13 + b12 - b10 + 2*b8 + b5 - b2 - b1 + 1) * q^91 + (2*b15 + 2*b14 + 2*b12 + 2*b11 - 2*b9 + 2*b8 + 3*b7 + 3*b5 + 2*b4 + 2*b3 - 3) * q^92 + (-2*b15 - 2*b14 + b11 - 7*b10 - 3*b7 - 4*b6 + b5 + b4 - 5*b1 + 5) * q^93 + (-b15 - b14 - 2*b13 + b11 - 7*b10 - 4*b9 - 6*b7 - 4*b6 + 2*b3 - 4*b2 - 4*b1 + 2) * q^94 + (2*b15 + 2*b14 + 2*b12 - b11 - 5*b10 - 2*b9 + 7*b8 - 2*b7 - b5 - 3*b4 + 2*b3 - 3*b2 - 1) * q^95 + (b15 - 2*b13 - 3*b12 + b11 + b10 - b9 + 2*b8 + 2*b6 - 4*b5 + 3*b2 + 4*b1 + 5) * q^96 + (b15 + 3*b14 - b13 - b12 + 4*b10 - b9 - 2*b8 + 3*b7 + 3*b6 + 3*b5 + 2*b4 - 2*b3 - 2*b1) * q^97 - b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 3 q^{2} - 2 q^{3} - 11 q^{4} - 5 q^{5} - 3 q^{6} + 4 q^{7} + 5 q^{8} - 12 q^{9}+O(q^{10})$$ 16 * q + 3 * q^2 - 2 * q^3 - 11 * q^4 - 5 * q^5 - 3 * q^6 + 4 * q^7 + 5 * q^8 - 12 * q^9 $$16 q + 3 q^{2} - 2 q^{3} - 11 q^{4} - 5 q^{5} - 3 q^{6} + 4 q^{7} + 5 q^{8} - 12 q^{9} - 12 q^{10} + 18 q^{12} + 7 q^{13} + 2 q^{14} - 18 q^{15} + 17 q^{16} + 5 q^{17} - 11 q^{18} - 19 q^{19} + q^{20} - 8 q^{21} + 32 q^{23} + 35 q^{24} + 7 q^{25} - 27 q^{26} + 10 q^{27} - 4 q^{28} - 3 q^{29} + 2 q^{30} - 7 q^{31} - 32 q^{32} - 24 q^{34} + 5 q^{35} + 52 q^{36} + 4 q^{37} - 5 q^{38} - 11 q^{39} + 10 q^{40} + 10 q^{41} + 3 q^{42} + 8 q^{43} + 70 q^{45} + 42 q^{46} - 23 q^{47} - 36 q^{48} - 4 q^{49} - 52 q^{50} + 29 q^{51} - 33 q^{52} + 4 q^{53} - 60 q^{54} + 11 q^{57} + 20 q^{58} + 17 q^{59} - 30 q^{60} + 7 q^{61} - 79 q^{62} + 2 q^{63} + 7 q^{64} + 8 q^{65} - 38 q^{67} + 2 q^{68} + 10 q^{69} - 18 q^{70} - 14 q^{71} + 35 q^{73} + 29 q^{74} + 9 q^{75} - 52 q^{76} - 58 q^{78} - 15 q^{79} - 87 q^{80} - 14 q^{81} + 19 q^{82} - 5 q^{83} - 8 q^{84} - 6 q^{85} - 52 q^{86} + 72 q^{87} + 74 q^{89} + 14 q^{90} + 13 q^{91} - 55 q^{92} + 32 q^{93} + 24 q^{94} - 32 q^{95} + 42 q^{96} + 20 q^{97} - 2 q^{98}+O(q^{100})$$ 16 * q + 3 * q^2 - 2 * q^3 - 11 * q^4 - 5 * q^5 - 3 * q^6 + 4 * q^7 + 5 * q^8 - 12 * q^9 - 12 * q^10 + 18 * q^12 + 7 * q^13 + 2 * q^14 - 18 * q^15 + 17 * q^16 + 5 * q^17 - 11 * q^18 - 19 * q^19 + q^20 - 8 * q^21 + 32 * q^23 + 35 * q^24 + 7 * q^25 - 27 * q^26 + 10 * q^27 - 4 * q^28 - 3 * q^29 + 2 * q^30 - 7 * q^31 - 32 * q^32 - 24 * q^34 + 5 * q^35 + 52 * q^36 + 4 * q^37 - 5 * q^38 - 11 * q^39 + 10 * q^40 + 10 * q^41 + 3 * q^42 + 8 * q^43 + 70 * q^45 + 42 * q^46 - 23 * q^47 - 36 * q^48 - 4 * q^49 - 52 * q^50 + 29 * q^51 - 33 * q^52 + 4 * q^53 - 60 * q^54 + 11 * q^57 + 20 * q^58 + 17 * q^59 - 30 * q^60 + 7 * q^61 - 79 * q^62 + 2 * q^63 + 7 * q^64 + 8 * q^65 - 38 * q^67 + 2 * q^68 + 10 * q^69 - 18 * q^70 - 14 * q^71 + 35 * q^73 + 29 * q^74 + 9 * q^75 - 52 * q^76 - 58 * q^78 - 15 * q^79 - 87 * q^80 - 14 * q^81 + 19 * q^82 - 5 * q^83 - 8 * q^84 - 6 * q^85 - 52 * q^86 + 72 * q^87 + 74 * q^89 + 14 * q^90 + 13 * q^91 - 55 * q^92 + 32 * q^93 + 24 * q^94 - 32 * q^95 + 42 * q^96 + 20 * q^97 - 2 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} - 1175 x^{7} + 2135 x^{6} - 2300 x^{5} + 1850 x^{4} - 925 x^{3} + 700 x^{2} + \cdots + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( - 33722118880975 \nu^{15} - 215936836813390 \nu^{14} + 154302737507927 \nu^{13} + \cdots + 88\!\cdots\!50 ) / 45\!\cdots\!60$$ (-33722118880975*v^15 - 215936836813390*v^14 + 154302737507927*v^13 - 3234633146529285*v^12 + 4494612950061958*v^11 - 20835947263746904*v^10 + 25956864481307657*v^9 - 65098048812157320*v^8 + 30869277708222543*v^7 - 160665000409949210*v^6 + 100524107363786610*v^5 - 560336279122558890*v^4 + 546963632352958070*v^3 - 434474896057227745*v^2 + 120133073008741205*v + 88528161377575350) / 454580475630153760 $$\beta_{2}$$ $$=$$ $$( - 686277383720070 \nu^{15} + \cdots - 15\!\cdots\!35 ) / 45\!\cdots\!60$$ (-686277383720070*v^15 - 1360333162376645*v^14 + 444742620384542*v^13 - 24553779304461089*v^12 + 45808084826141647*v^11 - 179218800902324756*v^10 + 292393929070812298*v^9 - 592228758909440833*v^8 + 274072637813433960*v^7 - 1226946693965489055*v^6 + 2313198962618101340*v^5 - 5164313069616781590*v^4 + 5505326380247646890*v^3 - 4265354995237286060*v^2 + 1164454349991851775*v - 1569079812936964435) / 454580475630153760 $$\beta_{3}$$ $$=$$ $$( - 874936745197272 \nu^{15} - 514706296938285 \nu^{14} + \cdots - 92\!\cdots\!35 ) / 45\!\cdots\!60$$ (-874936745197272*v^15 - 514706296938285*v^14 - 5527222230184928*v^13 - 8597109141630243*v^12 - 9687442744709873*v^11 - 69609203132183412*v^10 + 41469078152400808*v^9 - 248365491806816213*v^8 - 292096370375405502*v^7 - 583922916320059255*v^6 + 395306407846606840*v^5 - 2070346335621533450*v^4 + 1021707328353270230*v^3 - 1173049633586345730*v^2 + 368306587037923705*v - 928639412535219835) / 454580475630153760 $$\beta_{4}$$ $$=$$ $$( - 15\!\cdots\!39 \nu^{15} + \cdots + 95\!\cdots\!50 ) / 45\!\cdots\!60$$ (-1565624412275039*v^15 + 4832412997019746*v^14 - 21672367495495401*v^13 + 49577176484430299*v^12 - 129795369286809034*v^11 + 214474805906670840*v^10 - 347621794857624183*v^9 + 312652728515985000*v^8 - 887685879699924177*v^7 + 1770758763681867190*v^6 - 3159142418183242750*v^5 + 2957115228664641670*v^4 - 1979053855141622330*v^3 + 366777001484809055*v^2 - 739274040704445675*v + 95214593589332550) / 454580475630153760 $$\beta_{5}$$ $$=$$ $$( 16\!\cdots\!90 \nu^{15} + \cdots - 18\!\cdots\!25 ) / 45\!\cdots\!60$$ (1676337845170790*v^15 - 5194753922189191*v^14 + 23996414540642882*v^13 - 55793399026072099*v^12 + 149577475159066613*v^11 - 254981761232364844*v^10 + 433123005758956374*v^9 - 436889234085257675*v^8 + 1101433166283370416*v^7 - 2030901870569891045*v^6 + 3769863137668677300*v^5 - 4085868252644216690*v^4 + 3585858111334770990*v^3 - 1476295694818421740*v^2 + 1479871527393288525*v - 189874590198959025) / 454580475630153760 $$\beta_{6}$$ $$=$$ $$( - 16\!\cdots\!90 \nu^{15} + \cdots + 18\!\cdots\!25 ) / 45\!\cdots\!60$$ (-1676337845170790*v^15 + 5194753922189191*v^14 - 23996414540642882*v^13 + 55793399026072099*v^12 - 149577475159066613*v^11 + 254981761232364844*v^10 - 433123005758956374*v^9 + 436889234085257675*v^8 - 1101433166283370416*v^7 + 2030901870569891045*v^6 - 3769863137668677300*v^5 + 4085868252644216690*v^4 - 3585858111334770990*v^3 + 1476295694818421740*v^2 - 1025291051763134765*v + 189874590198959025) / 454580475630153760 $$\beta_{7}$$ $$=$$ $$( - 35\!\cdots\!14 \nu^{15} + \cdots + 82\!\cdots\!05 ) / 45\!\cdots\!60$$ (-3541126455103014*v^15 + 10589657246428067*v^14 - 49791707208255586*v^13 + 113470349300804375*v^12 - 307771508285388489*v^11 + 517957948939998988*v^10 - 888411928763985334*v^9 + 893532845981546087*v^8 - 2331418980078086280*v^7 + 4191692862454263993*v^6 - 7720969982054884100*v^5 + 8245114954100718810*v^4 - 7111420221063134790*v^3 + 3822505603323246020*v^2 - 2913263414629337545*v + 828358364029344005) / 454580475630153760 $$\beta_{8}$$ $$=$$ $$( - 38\!\cdots\!02 \nu^{15} + \cdots + 22\!\cdots\!25 ) / 45\!\cdots\!60$$ (-3808583743573302*v^15 + 9860126818444867*v^14 - 48487759413006482*v^13 + 100202312298850263*v^12 - 277961025462873673*v^11 + 422449273531319756*v^10 - 718628211268788150*v^9 + 585481222317834807*v^8 - 2124840867370928280*v^7 + 3587400018998705673*v^6 - 6360567528847132580*v^5 + 5600600192035351850*v^4 - 4088764696945967030*v^3 + 1543886107663682020*v^2 - 2299231619016502345*v + 22442708010214725) / 454580475630153760 $$\beta_{9}$$ $$=$$ $$( 41\!\cdots\!60 \nu^{15} + \cdots - 29\!\cdots\!75 ) / 45\!\cdots\!60$$ (4147870816217860*v^15 - 9098179878471885*v^14 + 48642002600512068*v^13 - 88958591156351927*v^12 + 262233539631762351*v^11 - 349871456206210068*v^10 + 623455379026927044*v^9 - 381247122554708893*v^8 + 2150910817199786310*v^7 - 3112048315103081495*v^6 + 5535211606668989440*v^5 - 3712550967753264370*v^4 + 2767110524781279550*v^3 - 1335895153189756830*v^2 + 2975718288695695725*v - 29045982804284875) / 454580475630153760 $$\beta_{10}$$ $$=$$ $$( 73\!\cdots\!73 \nu^{15} + \cdots - 84\!\cdots\!55 ) / 45\!\cdots\!60$$ (7327526319488073*v^15 - 21838143406687493*v^14 + 102438964384476967*v^13 - 232311097915978782*v^12 + 627185674080861763*v^11 - 1047203823403574964*v^10 + 1775572940212630785*v^9 - 1735699601854553351*v^8 + 4630478387715251365*v^7 - 8426965416523262079*v^6 + 15544790585628961210*v^5 - 16343113922700919960*v^4 + 12987506946195918920*v^3 - 5718121221686276935*v^2 + 4454224626365266160*v - 845040850217512955) / 454580475630153760 $$\beta_{11}$$ $$=$$ $$( - 54\!\cdots\!19 \nu^{15} + \cdots + 86\!\cdots\!50 ) / 22\!\cdots\!80$$ (-5481281688546819*v^15 + 16493950936678582*v^14 - 76461534295292341*v^13 + 174584775915853027*v^12 - 466393479528723526*v^11 + 782325573976058616*v^10 - 1307780222870847083*v^9 + 1276995934654789444*v^8 - 3395227103875767205*v^7 + 6340299488823316290*v^6 - 11480502622702818310*v^5 + 12158198218393584390*v^4 - 9161804501671007210*v^3 + 3888216659798809835*v^2 - 2965788298433688075*v + 864066394282347650) / 227290237815076880 $$\beta_{12}$$ $$=$$ $$( - 11\!\cdots\!43 \nu^{15} + \cdots + 59\!\cdots\!75 ) / 45\!\cdots\!60$$ (-11668286039317543*v^15 + 32209452556198329*v^14 - 155422622027191961*v^13 + 335017774059801892*v^12 - 918061503999498391*v^11 + 1456783386223745044*v^10 - 2469198487594261567*v^9 + 2179375478325730279*v^8 - 6793193349504836855*v^7 + 11856118418058363915*v^6 - 21771993760224740470*v^5 + 21003963276182506660*v^4 - 15299497235328621900*v^3 + 5270338292583504165*v^2 - 5226852994856502950*v + 59418569180970775) / 454580475630153760 $$\beta_{13}$$ $$=$$ $$( - 16\!\cdots\!01 \nu^{15} + \cdots + 16\!\cdots\!55 ) / 45\!\cdots\!60$$ (-16296167766296901*v^15 + 46657645630438849*v^14 - 220693813391828059*v^13 + 490255203026234886*v^12 - 1324081568271118439*v^11 + 2171662124721910708*v^10 - 3646287371621077981*v^9 + 3462410165728884427*v^8 - 9848220683260194529*v^7 + 17797250626139187779*v^6 - 31784433905471092610*v^5 + 32902654497697219800*v^4 - 24876887445936951080*v^3 + 11522918100224582795*v^2 - 8819030598088243120*v + 1667167131638041055) / 454580475630153760 $$\beta_{14}$$ $$=$$ $$( 16\!\cdots\!14 \nu^{15} + \cdots - 10\!\cdots\!35 ) / 45\!\cdots\!60$$ (16576781827677814*v^15 - 45453688208119129*v^14 + 219284175557671186*v^13 - 470575312000685073*v^12 + 1289205279920317419*v^11 - 2038073676823909444*v^10 + 3445236405584454006*v^9 - 3032211154151816061*v^8 + 9585093068819660548*v^7 - 16816185325336642299*v^6 + 30431971572347844100*v^5 - 29120858031490839590*v^4 + 20842243045981534410*v^3 - 8417551544734639360*v^2 + 7973236770827366535*v - 1025213227739747135) / 454580475630153760 $$\beta_{15}$$ $$=$$ $$( - 31\!\cdots\!64 \nu^{15} + \cdots + 34\!\cdots\!95 ) / 45\!\cdots\!60$$ (-31742657223153564*v^15 + 90930749048576029*v^14 - 431347940936808556*v^13 + 955931937354509231*v^12 - 2590293343054165503*v^11 + 4236554416589668948*v^10 - 7143632052250437628*v^9 + 6742540241983255677*v^8 - 19234203982822940782*v^7 + 34610408690995495111*v^6 - 62595008084751532400*v^5 + 64110835668533354210*v^4 - 48322167102593217550*v^3 + 21980961042064417270*v^2 - 18197077383887239845*v + 3492178872565443195) / 454580475630153760
 $$\nu$$ $$=$$ $$\beta_{6} + \beta_{5}$$ b6 + b5 $$\nu^{2}$$ $$=$$ $$\beta_{11} + 2\beta_{10} - 2\beta_{8} + 3\beta_{7} + \beta_{2} - 2$$ b11 + 2*b10 - 2*b8 + 3*b7 + b2 - 2 $$\nu^{3}$$ $$=$$ $$-\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{8} - 5\beta_{6} - \beta_{5} - \beta_{2} + \beta_1$$ -b15 + b13 + b12 - b10 + b8 - 5*b6 - b5 - b2 + b1 $$\nu^{4}$$ $$=$$ $$- \beta_{15} - \beta_{14} - \beta_{11} - 8 \beta_{10} - 6 \beta_{9} - 15 \beta_{7} + \beta_{5} + \beta_{4} - 6 \beta_{2} - \beta _1 + 1$$ -b15 - b14 - b11 - 8*b10 - 6*b9 - 15*b7 + b5 + b4 - 6*b2 - b1 + 1 $$\nu^{5}$$ $$=$$ $$- 7 \beta_{14} - 7 \beta_{13} - 7 \beta_{12} + 7 \beta_{11} - 8 \beta_{8} + 8 \beta_{7} + 10 \beta_{6} - 10 \beta_{4} - \beta_{3} + 8 \beta_{2} - 18 \beta _1 - 3$$ -7*b14 - 7*b13 - 7*b12 + 7*b11 - 8*b8 + 8*b7 + 10*b6 - 10*b4 - b3 + 8*b2 - 18*b1 - 3 $$\nu^{6}$$ $$=$$ $$7 \beta_{15} + \beta_{14} - 7 \beta_{13} + 11 \beta_{12} + 54 \beta_{10} + 35 \beta_{9} + 40 \beta_{8} + 47 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} + 2 \beta_1$$ 7*b15 + b14 - 7*b13 + 11*b12 + 54*b10 + 35*b9 + 40*b8 + 47*b7 - 10*b6 - 10*b5 - 2*b4 + 6*b3 + 2*b1 $$\nu^{7}$$ $$=$$ $$42 \beta_{15} + 54 \beta_{14} + 13 \beta_{12} - 45 \beta_{11} + 12 \beta_{10} - 13 \beta_{9} + 30 \beta_{8} - 86 \beta_{7} + 76 \beta_{5} + 165 \beta_{4} + 42 \beta_{3} - 58 \beta_{2} + 28$$ 42*b15 + 54*b14 + 13*b12 - 45*b11 + 12*b10 - 13*b9 + 30*b8 - 86*b7 + 76*b5 + 165*b4 + 42*b3 - 58*b2 + 28 $$\nu^{8}$$ $$=$$ $$- 9 \beta_{15} + 38 \beta_{13} - 210 \beta_{12} + 89 \beta_{11} - 234 \beta_{10} - 89 \beta_{9} - 308 \beta_{8} + 80 \beta_{6} + 28 \beta_{5} + 210 \beta_{2} - 28 \beta _1 - 98$$ -9*b15 + 38*b13 - 210*b12 + 89*b11 - 234*b10 - 89*b9 - 308*b8 + 80*b6 + 28*b5 + 210*b2 - 28*b1 - 98 $$\nu^{9}$$ $$=$$ $$- 346 \beta_{15} - 346 \beta_{14} + 248 \beta_{13} + 117 \beta_{11} + 38 \beta_{10} + 290 \beta_{9} + 614 \beta_{7} - 524 \beta_{6} - 1000 \beta_{5} - 1000 \beta_{4} - 248 \beta_{3} + 290 \beta_{2} + 476 \beta _1 - 94$$ -346*b15 - 346*b14 + 248*b13 + 117*b11 + 38*b10 + 290*b9 + 614*b7 - 524*b6 - 1000*b5 - 1000*b4 - 248*b3 + 290*b2 + 476*b1 - 94 $$\nu^{10}$$ $$=$$ $$- 56 \beta_{14} - 56 \beta_{13} + 1290 \beta_{12} - 1290 \beta_{11} + 1992 \beta_{8} - 1992 \beta_{7} - 272 \beta_{6} + 272 \beta_{4} - 131 \beta_{3} - 1931 \beta_{2} + 328 \beta _1 + 1406$$ -56*b14 - 56*b13 + 1290*b12 - 1290*b11 + 1992*b8 - 1992*b7 - 272*b6 + 272*b4 - 131*b3 - 1931*b2 + 328*b1 + 1406 $$\nu^{11}$$ $$=$$ $$2174 \beta_{15} + 1477 \beta_{14} - 2174 \beta_{13} - 913 \beta_{12} - 447 \beta_{10} - 1890 \beta_{9} - 1625 \beta_{8} - 2621 \beta_{7} + 6160 \beta_{6} + 6160 \beta_{5} + 3461 \beta_{4} + 697 \beta_{3} + \cdots - 3461 \beta_1$$ 2174*b15 + 1477*b14 - 2174*b13 - 913*b12 - 447*b10 - 1890*b9 - 1625*b8 - 2621*b7 + 6160*b6 + 6160*b5 + 3461*b4 + 697*b3 - 3461*b1 $$\nu^{12}$$ $$=$$ $$284 \beta_{15} + 848 \beta_{14} - 4374 \beta_{12} + 8050 \beta_{11} + 8579 \beta_{10} + 4374 \beta_{9} - 8295 \beta_{8} + 21093 \beta_{7} - 2273 \beta_{5} - 4375 \beta_{4} + 284 \beta_{3} + 12424 \beta_{2} + \cdots - 8669$$ 284*b15 + 848*b14 - 4374*b12 + 8050*b11 + 8579*b10 + 4374*b9 - 8295*b8 + 21093*b7 - 2273*b5 - 4375*b4 + 284*b3 + 12424*b2 - 8669 $$\nu^{13}$$ $$=$$ $$- 8898 \beta_{15} + 13556 \beta_{13} + 12425 \beta_{12} - 6647 \beta_{11} + 2237 \beta_{10} + 6647 \beta_{9} + 17819 \beta_{8} - 38325 \beta_{6} - 22394 \beta_{5} - 12425 \beta_{2} + 22394 \beta _1 + 5394$$ -8898*b15 + 13556*b13 + 12425*b12 - 6647*b11 + 2237*b10 + 6647*b9 + 17819*b8 - 38325*b6 - 22394*b5 - 12425*b2 + 22394*b1 + 5394 $$\nu^{14}$$ $$=$$ $$- 3382 \beta_{15} - 3382 \beta_{14} + 1131 \beta_{13} - 29041 \beta_{11} - 85364 \beta_{10} - 50750 \beta_{9} - 133987 \beta_{7} + 17557 \beta_{6} + 31350 \beta_{5} + 31350 \beta_{4} - 1131 \beta_{3} + \cdots + 31232$$ -3382*b15 - 3382*b14 + 1131*b13 - 29041*b11 - 85364*b10 - 50750*b9 - 133987*b7 + 17557*b6 + 31350*b5 + 31350*b4 - 1131*b3 - 50750*b2 - 13793*b1 + 31232 $$\nu^{15}$$ $$=$$ $$- 54132 \beta_{14} - 54132 \beta_{13} - 82100 \beta_{12} + 82100 \beta_{11} - 120596 \beta_{8} + 120596 \beta_{7} + 143446 \beta_{6} - 143446 \beta_{4} - 30172 \beta_{3} + 128698 \beta_{2} + \cdots - 67213$$ -54132*b14 - 54132*b13 - 82100*b12 + 82100*b11 - 120596*b8 + 120596*b7 + 143446*b6 - 143446*b4 - 30172*b3 + 128698*b2 - 96554*b1 - 67213

## Character values

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

 $$n$$ $$122$$ $$365$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{7} - \beta_{8} + \beta_{10}$$

## 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
 −0.788594 − 2.42704i −0.206962 − 0.636964i 0.435488 + 1.34029i 0.751051 + 2.31150i −1.38112 − 1.00344i 0.183009 + 0.132964i 0.901622 + 0.655067i 1.60551 + 1.16647i −0.788594 + 2.42704i −0.206962 + 0.636964i 0.435488 − 1.34029i 0.751051 − 2.31150i −1.38112 + 1.00344i 0.183009 − 0.132964i 0.901622 − 0.655067i 1.60551 − 1.16647i
−0.788594 2.42704i 0.332181 + 0.241344i −3.65062 + 2.65233i 1.05961 3.26115i 0.323795 0.996539i 0.809017 0.587785i 5.18703 + 3.76860i −0.874954 2.69283i −8.75055
148.2 −0.206962 0.636964i −2.54013 1.84551i 1.25514 0.911915i 0.662464 2.03885i −0.649815 + 1.99992i 0.809017 0.587785i −1.92429 1.39808i 2.11929 + 6.52251i −1.43578
148.3 0.435488 + 1.34029i 1.75021 + 1.27160i 0.0112975 0.00820814i −0.565930 + 1.74175i −0.942126 + 2.89957i 0.809017 0.587785i 2.29616 + 1.66826i 0.519216 + 1.59798i −2.58091
148.4 0.751051 + 2.31150i −1.16030 0.843005i −3.16091 + 2.29654i 0.388938 1.19703i 1.07716 3.31516i 0.809017 0.587785i −3.74989 2.72445i −0.291419 0.896896i 3.05904
323.1 −1.38112 1.00344i 0.708129 2.17940i 0.282562 + 0.869638i −3.28976 + 2.39015i −3.16491 + 2.29944i −0.309017 0.951057i −0.572703 + 1.76260i −1.82128 1.32323i 6.94194
323.2 0.183009 + 0.132964i −0.0677147 + 0.208405i −0.602221 1.85345i 2.01892 1.46683i −0.0401026 + 0.0291363i −0.309017 0.951057i 0.276036 0.849550i 2.38820 + 1.73513i 0.564516
323.3 0.901622 + 0.655067i −0.883423 + 2.71890i −0.234224 0.720867i −2.79603 + 2.03143i −2.57757 + 1.87272i −0.309017 0.951057i 0.949813 2.92322i −4.18492 3.04052i −3.85168
323.4 1.60551 + 1.16647i 0.861043 2.65002i 0.598967 + 1.84343i 0.0217822 0.0158257i 4.47357 3.25024i −0.309017 0.951057i 0.0378378 0.116453i −3.85415 2.80020i 0.0534317
372.1 −0.788594 + 2.42704i 0.332181 0.241344i −3.65062 2.65233i 1.05961 + 3.26115i 0.323795 + 0.996539i 0.809017 + 0.587785i 5.18703 3.76860i −0.874954 + 2.69283i −8.75055
372.2 −0.206962 + 0.636964i −2.54013 + 1.84551i 1.25514 + 0.911915i 0.662464 + 2.03885i −0.649815 1.99992i 0.809017 + 0.587785i −1.92429 + 1.39808i 2.11929 6.52251i −1.43578
372.3 0.435488 1.34029i 1.75021 1.27160i 0.0112975 + 0.00820814i −0.565930 1.74175i −0.942126 2.89957i 0.809017 + 0.587785i 2.29616 1.66826i 0.519216 1.59798i −2.58091
372.4 0.751051 2.31150i −1.16030 + 0.843005i −3.16091 2.29654i 0.388938 + 1.19703i 1.07716 + 3.31516i 0.809017 + 0.587785i −3.74989 + 2.72445i −0.291419 + 0.896896i 3.05904
729.1 −1.38112 + 1.00344i 0.708129 + 2.17940i 0.282562 0.869638i −3.28976 2.39015i −3.16491 2.29944i −0.309017 + 0.951057i −0.572703 1.76260i −1.82128 + 1.32323i 6.94194
729.2 0.183009 0.132964i −0.0677147 0.208405i −0.602221 + 1.85345i 2.01892 + 1.46683i −0.0401026 0.0291363i −0.309017 + 0.951057i 0.276036 + 0.849550i 2.38820 1.73513i 0.564516
729.3 0.901622 0.655067i −0.883423 2.71890i −0.234224 + 0.720867i −2.79603 2.03143i −2.57757 1.87272i −0.309017 + 0.951057i 0.949813 + 2.92322i −4.18492 + 3.04052i −3.85168
729.4 1.60551 1.16647i 0.861043 + 2.65002i 0.598967 1.84343i 0.0217822 + 0.0158257i 4.47357 + 3.25024i −0.309017 + 0.951057i 0.0378378 + 0.116453i −3.85415 + 2.80020i 0.0534317
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 729.4 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.x 16
11.b odd 2 1 77.2.f.b 16
11.c even 5 1 847.2.a.o 8
11.c even 5 2 847.2.f.v 16
11.c even 5 1 inner 847.2.f.x 16
11.d odd 10 1 77.2.f.b 16
11.d odd 10 1 847.2.a.p 8
11.d odd 10 2 847.2.f.w 16
33.d even 2 1 693.2.m.i 16
33.f even 10 1 693.2.m.i 16
33.f even 10 1 7623.2.a.ct 8
33.h odd 10 1 7623.2.a.cw 8
77.b even 2 1 539.2.f.e 16
77.h odd 6 2 539.2.q.g 32
77.i even 6 2 539.2.q.f 32
77.j odd 10 1 5929.2.a.bs 8
77.l even 10 1 539.2.f.e 16
77.l even 10 1 5929.2.a.bt 8
77.n even 30 2 539.2.q.f 32
77.o odd 30 2 539.2.q.g 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 11.b odd 2 1
77.2.f.b 16 11.d odd 10 1
539.2.f.e 16 77.b even 2 1
539.2.f.e 16 77.l even 10 1
539.2.q.f 32 77.i even 6 2
539.2.q.f 32 77.n even 30 2
539.2.q.g 32 77.h odd 6 2
539.2.q.g 32 77.o odd 30 2
693.2.m.i 16 33.d even 2 1
693.2.m.i 16 33.f even 10 1
847.2.a.o 8 11.c even 5 1
847.2.a.p 8 11.d odd 10 1
847.2.f.v 16 11.c even 5 2
847.2.f.w 16 11.d odd 10 2
847.2.f.x 16 1.a even 1 1 trivial
847.2.f.x 16 11.c even 5 1 inner
5929.2.a.bs 8 77.j odd 10 1
5929.2.a.bt 8 77.l even 10 1
7623.2.a.ct 8 33.f even 10 1
7623.2.a.cw 8 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 $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}^{16} - 3 T_{2}^{15} + 14 T_{2}^{14} - 32 T_{2}^{13} + 86 T_{2}^{12} - 145 T_{2}^{11} + 245 T_{2}^{10} - 245 T_{2}^{9} + 640 T_{2}^{8} - 1175 T_{2}^{7} + 2135 T_{2}^{6} - 2300 T_{2}^{5} + 1850 T_{2}^{4} - 925 T_{2}^{3} + 700 T_{2}^{2} + \cdots + 25$$ T2^16 - 3*T2^15 + 14*T2^14 - 32*T2^13 + 86*T2^12 - 145*T2^11 + 245*T2^10 - 245*T2^9 + 640*T2^8 - 1175*T2^7 + 2135*T2^6 - 2300*T2^5 + 1850*T2^4 - 925*T2^3 + 700*T2^2 - 200*T2 + 25 $$T_{3}^{16} + 2 T_{3}^{15} + 14 T_{3}^{14} + 26 T_{3}^{13} + 124 T_{3}^{12} + 100 T_{3}^{11} + 747 T_{3}^{10} - 178 T_{3}^{9} + 4253 T_{3}^{8} + 872 T_{3}^{7} + 8452 T_{3}^{6} + 14920 T_{3}^{5} + 22464 T_{3}^{4} - 14304 T_{3}^{3} + \cdots + 256$$ T3^16 + 2*T3^15 + 14*T3^14 + 26*T3^13 + 124*T3^12 + 100*T3^11 + 747*T3^10 - 178*T3^9 + 4253*T3^8 + 872*T3^7 + 8452*T3^6 + 14920*T3^5 + 22464*T3^4 - 14304*T3^3 + 3904*T3^2 - 128*T3 + 256 $$T_{13}^{16} - 7 T_{13}^{15} + 39 T_{13}^{14} - 81 T_{13}^{13} + 484 T_{13}^{12} - 2465 T_{13}^{11} + 25007 T_{13}^{10} - 87462 T_{13}^{9} + 357453 T_{13}^{8} - 687552 T_{13}^{7} + 2166132 T_{13}^{6} - 2287000 T_{13}^{5} + \cdots + 188897536$$ T13^16 - 7*T13^15 + 39*T13^14 - 81*T13^13 + 484*T13^12 - 2465*T13^11 + 25007*T13^10 - 87462*T13^9 + 357453*T13^8 - 687552*T13^7 + 2166132*T13^6 - 2287000*T13^5 + 10501024*T13^4 + 1175584*T13^3 + 54886464*T13^2 - 11105152*T13 + 188897536

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 3 T^{15} + 14 T^{14} - 32 T^{13} + \cdots + 25$$
$3$ $$T^{16} + 2 T^{15} + 14 T^{14} + 26 T^{13} + \cdots + 256$$
$5$ $$T^{16} + 5 T^{15} + 19 T^{14} + 59 T^{13} + \cdots + 256$$
$7$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{4}$$
$11$ $$T^{16}$$
$13$ $$T^{16} - 7 T^{15} + 39 T^{14} + \cdots + 188897536$$
$17$ $$T^{16} - 5 T^{15} + 81 T^{14} + \cdots + 2611456$$
$19$ $$T^{16} + 19 T^{15} + 189 T^{14} + \cdots + 62726400$$
$23$ $$(T^{8} - 16 T^{7} + 40 T^{6} + 342 T^{5} + \cdots + 859)^{2}$$
$29$ $$T^{16} + 3 T^{15} + 116 T^{14} + \cdots + 245025$$
$31$ $$T^{16} + 7 T^{15} + \cdots + 2629638400$$
$37$ $$T^{16} - 4 T^{15} + 84 T^{14} + \cdots + 212521$$
$41$ $$T^{16} - 10 T^{15} + 196 T^{14} + \cdots + 13424896$$
$43$ $$(T^{8} - 4 T^{7} - 105 T^{6} + 268 T^{5} + \cdots - 971)^{2}$$
$47$ $$T^{16} + 23 T^{15} + \cdots + 5345713926400$$
$53$ $$T^{16} - 4 T^{15} + \cdots + 310840815961$$
$59$ $$T^{16} - 17 T^{15} + \cdots + 187142400$$
$61$ $$T^{16} - 7 T^{15} + 224 T^{14} + \cdots + 253446400$$
$67$ $$(T^{8} + 19 T^{7} - 16 T^{6} - 1160 T^{5} + \cdots - 27395)^{2}$$
$71$ $$T^{16} + 14 T^{15} + \cdots + 379119841$$
$73$ $$T^{16} - 35 T^{15} + \cdots + 105069332736$$
$79$ $$T^{16} + 15 T^{15} + \cdots + 15858514175625$$
$83$ $$T^{16} + 5 T^{15} + 14 T^{14} + \cdots + 756470016$$
$89$ $$(T^{8} - 37 T^{7} + 320 T^{6} + \cdots + 952400)^{2}$$
$97$ $$T^{16} - 20 T^{15} + \cdots + 4647025244416$$