# Properties

 Label 847.2.f.v 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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} + ( - \beta_{15} - \beta_{14} + \beta_{11} - \beta_{10}) q^{3} + (\beta_{10} + \beta_{9} + \beta_{7}) q^{4} + (\beta_{15} + 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{3} - 1) q^{5} + (\beta_{15} - \beta_{12} + 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{6} - \beta_{8} q^{7} + ( - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{8} + ( - \beta_{15} + \beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{2} - 1) q^{9}+O(q^{10})$$ q - b6 * q^2 + (-b15 - b14 + b11 - b10) * q^3 + (b10 + b9 + b7) * q^4 + (b15 + 2*b10 - b8 + b7 - b5 + b3 - 1) * q^5 + (b15 - b12 + 2*b10 + b9 - b8 - b5 + b4 + b3 + b2 + 1) * q^6 - b8 * q^7 + (-b15 - b14 + b13 + b9 + b7 - b6 - b5 - b4 - b3 + b2) * q^8 + (-b15 + b12 - b11 - 2*b10 + b9 - b2 - 1) * q^9 $$q - \beta_{6} q^{2} + ( - \beta_{15} - \beta_{14} + \beta_{11} - \beta_{10}) q^{3} + (\beta_{10} + \beta_{9} + \beta_{7}) q^{4} + (\beta_{15} + 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{3} - 1) q^{5} + (\beta_{15} - \beta_{12} + 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{6} - \beta_{8} q^{7} + ( - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{8} + ( - \beta_{15} + \beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{2} - 1) q^{9} + ( - 2 \beta_{8} + 2 \beta_{7} - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{10} + (\beta_{12} - \beta_{11} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} - \beta_{2} - 1) q^{12} + ( - \beta_{13} - \beta_{12} + \beta_{10} - 2 \beta_{8} - \beta_{5} + \beta_{2} + \beta_1 - 1) q^{13} + (\beta_{5} + \beta_{4} - \beta_1) q^{14} + (\beta_{15} - \beta_{14} - \beta_{13} - 2 \beta_{12} + 2 \beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{15}+ \cdots - \beta_1 q^{98}+O(q^{100})$$ q - b6 * q^2 + (-b15 - b14 + b11 - b10) * q^3 + (b10 + b9 + b7) * q^4 + (b15 + 2*b10 - b8 + b7 - b5 + b3 - 1) * q^5 + (b15 - b12 + 2*b10 + b9 - b8 - b5 + b4 + b3 + b2 + 1) * q^6 - b8 * q^7 + (-b15 - b14 + b13 + b9 + b7 - b6 - b5 - b4 - b3 + b2) * q^8 + (-b15 + b12 - b11 - 2*b10 + b9 - b2 - 1) * q^9 + (-2*b8 + 2*b7 - b3 + b2 - 2*b1 - 1) * q^10 + (b12 - b11 - b8 + b7 + b6 - b4 - b2 - 1) * q^12 + (-b13 - b12 + b10 - 2*b8 - b5 + b2 + b1 - 1) * q^13 + (b5 + b4 - b1) * q^14 + (b15 - b14 - b13 - 2*b12 + 2*b10 + b8 + b7 + b6 + b5 - b4 + 2*b3 + b1) * q^15 + (b14 - b12 + b9 + b7 - b4 + b2) * q^16 + (-b15 - b14 - b8 + 2*b7 - b5 - b4 - b3 - 2) * q^17 + (-b15 - b14 + b13 + b12 + b9 + b7 - 2*b4 + 2*b1) * q^18 + (-b13 - b10 + b9 - 3*b7 + b5 + b4 + b3 + b2 - b1 + 2) * q^19 + (b15 - 2*b12 + b11 - 2*b10 - b9 - b8 + b6 + b5 + 2*b2 - b1 + 1) * 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 + b13 - b11 + 2*b10 + b9 - 4*b8 - b5 + b1 - 4) * q^24 + (b15 + b14 + b13 - b11 + 3*b10 - 2*b7 + 3*b6 - b3 + 3*b1 - 2) * q^25 + (b12 + 2*b10 - b9 - b8 + 2*b7 + 2*b6 + 2*b5 + b4 - b1) * q^26 + (-b15 + b14 + 2*b12 - b11 - b10 - 2*b9 + 2*b7 - 2*b4 - b3 - 3*b2 - 5) * q^27 + (-b11 - b7 - b2) * q^28 + (-b14 - 2*b12 + 2*b10 + b9 - 3*b8 + 2*b7 - b6 - b5 + b3) * q^29 + (2*b15 + 2*b14 + b13 - b11 + 4*b10 - b9 - 4*b7 + 2*b6 - 2*b5 - 2*b4 - b3 - b2 + 4*b1 - 3) * q^30 + (2*b13 + b12 - b10 + b8 - b6 - 3*b5 - b2 + 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 + (b15 + 2*b10 - b5 + b1) * q^35 + (-2*b13 + b11 + 4*b10 - b9 + 4*b7 - 2*b6 - b5 - b4 + 2*b3 - b2 - b1 - 5) * q^36 + (-2*b15 + b14 + 2*b13 + b12 - 2*b10 + b8 - b6 - b5 - b4 - 3*b3 + b1) * q^37 + (-b15 - 2*b14 - 2*b10 + b8 - b7 + b5 - b3 + 1) * q^38 + (-b15 + 2*b14 + b12 + 2*b11 - 4*b10 - b9 + 3*b8 - 4*b7 + 3*b5 - b4 - b3 + b2 + 5) * q^39 + (2*b14 - b10 - 3*b9 - b8 - b7 + 3*b6 + 3*b5 + 3*b4 - 2*b3 - 3*b1) * q^40 + (b15 + b14 + b11 + b10 - 2*b9 - 2*b2 - 2) * q^41 + (b15 - b11 + 2*b10 + b9 - b8 + b6 - b5 + b1 - 1) * 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 + (-2*b15 + b13 + b12 + 2*b11 - b10 - 2*b9 - 2*b8 - 3*b6 - b2 - 3) * q^46 + (b13 - 2*b11 + 2*b9 - 3*b7 - b6 - 2*b5 - 2*b4 - b3 + 2*b2 + b1 + 2) * q^47 + (-b15 + b14 + b13 + 2*b12 + 2*b10 - 2*b9 - b8 + 3*b7 + b6 + b5 + b4 - 2*b3 - b1) * q^48 + (b10 - b8 + b7 - 1) * q^49 + (-b15 + b14 + 4*b12 - 4*b9 - b8 - 4*b7 - b5 + 2*b4 - b3 - 4*b2) * q^50 + (b15 + 2*b14 - b13 - 2*b10 - 2*b9 - 3*b7 - b6 - b5 - 2*b4 - b3 + 2*b1) * q^51 + (-2*b15 - 2*b14 - b13 - 3*b10 - b9 - 5*b7 - 2*b6 + b5 + b4 + b3 - b2 - 3*b1) * q^52 + (-2*b13 + 3*b11 - 5*b10 - 3*b9 - 2*b8 + b6 - 2) * 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*b13 + 4*b12 - b11 - 2*b10 + b9 + 2*b6 + b5 - 4*b2 - b1 - 4) * q^57 + (2*b15 + 2*b14 + b13 - 2*b11 + 4*b10 + 2*b9 + b7 + 2*b6 - b3 + 2*b2 + 2*b1) * q^58 + (b15 - b14 - b13 - b12 + b9 - b7 - 3*b6 - 3*b5 + 2*b3) * q^59 + (b15 - b14 - b12 + b11 + 3*b10 + b9 - 2*b8 - b7 + 3*b4 + b3 + 2*b2 + 3) * q^60 + (-b14 - b12 - b11 - b10 + b9 + b8 - 2*b7 + b5 + 3*b4 + 2) * q^61 + (b15 + b14 - b13 + 3*b12 + 7*b10 + 2*b9 - b8 + 6*b7 + b4 - b1) * q^62 + (-b13 - b11 - b9 + b7 + b3 - b2 - 1) * q^63 + (-b15 - 3*b13 + b12 + b11 - b10 - b9 - b8 + 2*b5 - b2 - 2*b1 - 2) * 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 + (b15 - 3*b13 - b12 + b11 + b10 - b9 - b8 + 5*b6 + 4*b5 + b2 - 4*b1) * q^68 + (-b13 + b11 - 3*b9 - 4*b6 + b3 - 3*b2 - 4*b1 - 3) * q^69 + (-b15 + b13 + b12 + b10 + 2*b7 - 2*b6 - 2*b5 - b3) * q^70 + (-3*b15 - 3*b14 - b12 + 3*b11 + b9 - 3*b8 + 3*b7 + b4 - 3*b3 + 4*b2 + 1) * q^71 + (3*b15 - b12 - 2*b11 + 6*b10 + b9 - 3*b8 + 5*b7 - 2*b5 - 4*b4 + 3*b3 - b2 - 6) * q^72 + (b15 - 2*b14 - b13 - 4*b10 + 2*b9 + 6*b8 - 5*b7 - b6 - b5 + 3*b3) * q^73 + (-2*b13 + 2*b11 + b10 + b9 + 4*b7 + 2*b3 + b2) * q^74 + (2*b15 - b12 - 2*b11 + b10 + 2*b9 + b8 - 3*b6 + 5*b5 + b2 - 5*b1 + 2) * 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*b13 + b12 + b11 - 2*b10 - b9 + b8 - 2*b6 + b5 - b2 - b1) * q^79 + (b15 + b14 - b13 - b11 - 9*b10 - 2*b9 - 5*b7 - b6 + 4*b5 + 4*b4 + b3 - 2*b2 - 5*b1 + 8) * q^80 + (3*b15 + b14 - 3*b13 + 3*b10 - 3*b9 + b8 - 4*b6 - 4*b5 + 2*b3) * q^81 + (b15 - b12 - 2*b11 + 2*b10 + b9 - b8 - 3*b5 + 3*b4 + b3 - b2 - 1) * q^82 + (2*b15 - 2*b14 - 2*b12 - b11 + 5*b10 + 2*b9 - 3*b8 + 4*b7 - b5 - b4 + 2*b3 + b2 - 3) * q^83 + (b10 - b9 - b8 + b7 - b6 - b5 - b4 + b1) * q^84 + (-b15 - b14 + b11 - 2*b10 - b9 + 3*b6 + 3*b5 + 3*b4 - b2) * q^85 + (-b15 + b13 - b12 - b11 - 3*b10 + b9 + 5*b8 - b6 + b5 + b2 - b1 + 6) * 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*b15 - 5*b12 - b11 + 3*b10 + b9 - 2*b8 + 2*b6 + 4*b5 + 5*b2 - 4*b1 + 3) * q^90 + (-b15 - b14 + b10 + b9 + b7 + b6 + b2 + b1 - 1) * q^91 + (2*b15 + 2*b14 - 2*b13 - 2*b12 + 7*b10 + 2*b9 - 2*b8 + 5*b7 + 2*b6 + 2*b5 + 3*b4 - 3*b1) * q^92 + (2*b14 + b12 - 2*b10 - b9 + 2*b8 + 3*b7 + 4*b5 - b4 - b2 - 4) * q^93 + (-2*b15 + b14 + b12 + 4*b11 - 2*b10 - b9 + 6*b7 + 4*b5 - 2*b3 + 3*b2 - 3) * q^94 + (2*b15 + 2*b14 - 2*b13 - 2*b12 + 7*b10 - b9 - 7*b8 + 5*b7 - 3*b6 - 3*b5 - b4 + b1) * q^95 + (2*b15 + 2*b14 - b13 - b11 + 4*b10 - 3*b9 + 2*b7 - 4*b6 + 2*b5 + 2*b4 + b3 - 3*b2 - 6*b1 - 5) * q^96 + (-3*b15 + b13 + b12 - b11 - b10 + b9 - 3*b8 - 3*b6 - 2*b5 - b2 + 2*b1 - 4) * q^97 - b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} + 2 q^{6} + 4 q^{7} - 5 q^{8} - 2 q^{9}+O(q^{10})$$ 16 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 5 * q^5 + 2 * q^6 + 4 * q^7 - 5 * q^8 - 2 * q^9 $$16 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} + 2 q^{6} + 4 q^{7} - 5 q^{8} - 2 q^{9} - 12 q^{10} + 18 q^{12} - 13 q^{13} - 3 q^{14} + 7 q^{15} - 18 q^{16} - 10 q^{17} + 19 q^{18} + 6 q^{19} - 24 q^{20} - 8 q^{21} + 32 q^{23} - 45 q^{24} - 23 q^{25} + 33 q^{26} - 20 q^{27} + 11 q^{28} + 12 q^{29} - 38 q^{30} - 2 q^{31} - 32 q^{32} - 24 q^{34} + 5 q^{35} - 38 q^{36} - 11 q^{37} + 15 q^{38} + 24 q^{39} - 5 q^{40} - 20 q^{41} - 2 q^{42} + 8 q^{43} + 70 q^{45} - 38 q^{46} + 7 q^{47} + 39 q^{48} - 4 q^{49} + 58 q^{50} - 16 q^{51} - 8 q^{52} - 41 q^{53} - 60 q^{54} - 9 q^{57} - 5 q^{58} - 18 q^{59} + 25 q^{60} + 12 q^{61} + 61 q^{62} + 12 q^{63} - 3 q^{64} + 8 q^{65} - 38 q^{67} + 7 q^{68} - 30 q^{69} + 12 q^{70} + q^{71} - 35 q^{72} - 60 q^{73} + 4 q^{74} + 4 q^{75} - 52 q^{76} - 58 q^{78} + 15 q^{79} + 83 q^{80} + 6 q^{81} - 6 q^{82} - 20 q^{83} + 17 q^{84} + 9 q^{85} + 48 q^{86} + 72 q^{87} + 74 q^{89} - 16 q^{90} - 7 q^{91} + 20 q^{92} - 53 q^{93} - 66 q^{94} + 53 q^{95} - 48 q^{96} - 35 q^{97} - 2 q^{98}+O(q^{100})$$ 16 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 5 * q^5 + 2 * q^6 + 4 * q^7 - 5 * q^8 - 2 * q^9 - 12 * q^10 + 18 * q^12 - 13 * q^13 - 3 * q^14 + 7 * q^15 - 18 * q^16 - 10 * q^17 + 19 * q^18 + 6 * q^19 - 24 * q^20 - 8 * q^21 + 32 * q^23 - 45 * q^24 - 23 * q^25 + 33 * q^26 - 20 * q^27 + 11 * q^28 + 12 * q^29 - 38 * q^30 - 2 * q^31 - 32 * q^32 - 24 * q^34 + 5 * q^35 - 38 * q^36 - 11 * q^37 + 15 * q^38 + 24 * q^39 - 5 * q^40 - 20 * q^41 - 2 * q^42 + 8 * q^43 + 70 * q^45 - 38 * q^46 + 7 * q^47 + 39 * q^48 - 4 * q^49 + 58 * q^50 - 16 * q^51 - 8 * q^52 - 41 * q^53 - 60 * q^54 - 9 * q^57 - 5 * q^58 - 18 * q^59 + 25 * q^60 + 12 * q^61 + 61 * q^62 + 12 * q^63 - 3 * q^64 + 8 * q^65 - 38 * q^67 + 7 * q^68 - 30 * q^69 + 12 * q^70 + q^71 - 35 * q^72 - 60 * q^73 + 4 * q^74 + 4 * q^75 - 52 * q^76 - 58 * q^78 + 15 * q^79 + 83 * q^80 + 6 * q^81 - 6 * q^82 - 20 * q^83 + 17 * q^84 + 9 * q^85 + 48 * q^86 + 72 * q^87 + 74 * q^89 - 16 * q^90 - 7 * q^91 + 20 * q^92 - 53 * q^93 - 66 * q^94 + 53 * q^95 - 48 * q^96 - 35 * 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 $$\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$365$$ $$\chi(n)$$ $$1$$ $$\beta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
148.1
 1.60551 − 1.16647i 0.901622 − 0.655067i 0.183009 − 0.132964i −1.38112 + 1.00344i 0.751051 + 2.31150i 0.435488 + 1.34029i −0.206962 − 0.636964i −0.788594 − 2.42704i 1.60551 + 1.16647i 0.901622 + 0.655067i 0.183009 + 0.132964i −1.38112 − 1.00344i 0.751051 − 2.31150i 0.435488 − 1.34029i −0.206962 + 0.636964i −0.788594 + 2.42704i
−0.613249 1.88739i −2.25424 1.63780i −1.56812 + 1.13930i −0.00832008 + 0.0256066i −1.70875 + 5.25900i 0.809017 0.587785i −0.0990607 0.0719718i 1.47215 + 4.53082i 0.0534317
148.2 −0.344389 1.05992i 2.31283 + 1.68037i 0.613206 0.445520i 1.06799 3.28693i 0.984546 3.03012i 0.809017 0.587785i −2.48664 1.80665i 1.59850 + 4.91966i −3.85168
148.3 −0.0699031 0.215140i 0.177280 + 0.128801i 1.57664 1.14549i −0.771159 + 2.37338i 0.0153178 0.0471435i 0.809017 0.587785i −0.722670 0.525051i −0.912213 2.80750i 0.564516
148.4 0.527541 + 1.62360i −1.85391 1.34694i −0.739758 + 0.537466i 1.25658 3.86735i 1.20889 3.72058i 0.809017 0.587785i 1.49936 + 1.08935i 0.695665 + 2.14104i 6.94194
323.1 −1.96628 1.42858i 0.443194 1.36401i 1.20736 + 3.71587i −1.01825 + 0.739805i −2.82005 + 2.04888i −0.309017 0.951057i 1.43233 4.40826i 0.762946 + 0.554312i 3.05904
323.2 −1.14012 0.828347i −0.668522 + 2.05750i −0.00431527 0.0132810i 1.48162 1.07646i 2.46652 1.79203i −0.309017 0.951057i −0.877057 + 2.69930i −1.35932 0.987607i −2.58091
323.3 0.541834 + 0.393666i 0.970243 2.98610i −0.479422 1.47551i −1.73435 + 1.26008i 1.70124 1.23602i −0.309017 0.951057i 0.735015 2.26214i −5.54838 4.03113i −1.43578
323.4 2.06456 + 1.49999i −0.126882 + 0.390502i 1.39441 + 4.29156i −2.77410 + 2.01550i −0.847707 + 0.615895i −0.309017 0.951057i −1.98127 + 6.09772i 2.29066 + 1.66426i −8.75055
372.1 −0.613249 + 1.88739i −2.25424 + 1.63780i −1.56812 1.13930i −0.00832008 0.0256066i −1.70875 5.25900i 0.809017 + 0.587785i −0.0990607 + 0.0719718i 1.47215 4.53082i 0.0534317
372.2 −0.344389 + 1.05992i 2.31283 1.68037i 0.613206 + 0.445520i 1.06799 + 3.28693i 0.984546 + 3.03012i 0.809017 + 0.587785i −2.48664 + 1.80665i 1.59850 4.91966i −3.85168
372.3 −0.0699031 + 0.215140i 0.177280 0.128801i 1.57664 + 1.14549i −0.771159 2.37338i 0.0153178 + 0.0471435i 0.809017 + 0.587785i −0.722670 + 0.525051i −0.912213 + 2.80750i 0.564516
372.4 0.527541 1.62360i −1.85391 + 1.34694i −0.739758 0.537466i 1.25658 + 3.86735i 1.20889 + 3.72058i 0.809017 + 0.587785i 1.49936 1.08935i 0.695665 2.14104i 6.94194
729.1 −1.96628 + 1.42858i 0.443194 + 1.36401i 1.20736 3.71587i −1.01825 0.739805i −2.82005 2.04888i −0.309017 + 0.951057i 1.43233 + 4.40826i 0.762946 0.554312i 3.05904
729.2 −1.14012 + 0.828347i −0.668522 2.05750i −0.00431527 + 0.0132810i 1.48162 + 1.07646i 2.46652 + 1.79203i −0.309017 + 0.951057i −0.877057 2.69930i −1.35932 + 0.987607i −2.58091
729.3 0.541834 0.393666i 0.970243 + 2.98610i −0.479422 + 1.47551i −1.73435 1.26008i 1.70124 + 1.23602i −0.309017 + 0.951057i 0.735015 + 2.26214i −5.54838 + 4.03113i −1.43578
729.4 2.06456 1.49999i −0.126882 0.390502i 1.39441 4.29156i −2.77410 2.01550i −0.847707 0.615895i −0.309017 + 0.951057i −1.98127 6.09772i 2.29066 1.66426i −8.75055
 $$n$$: 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.v 16
11.b odd 2 1 847.2.f.w 16
11.c even 5 1 847.2.a.o 8
11.c even 5 1 inner 847.2.f.v 16
11.c even 5 2 847.2.f.x 16
11.d odd 10 2 77.2.f.b 16
11.d odd 10 1 847.2.a.p 8
11.d odd 10 1 847.2.f.w 16
33.f even 10 2 693.2.m.i 16
33.f even 10 1 7623.2.a.ct 8
33.h odd 10 1 7623.2.a.cw 8
77.j odd 10 1 5929.2.a.bs 8
77.l even 10 2 539.2.f.e 16
77.l even 10 1 5929.2.a.bt 8
77.n even 30 4 539.2.q.f 32
77.o odd 30 4 539.2.q.g 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 11.d odd 10 2
539.2.f.e 16 77.l even 10 2
539.2.q.f 32 77.n even 30 4
539.2.q.g 32 77.o odd 30 4
693.2.m.i 16 33.f even 10 2
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 1.a even 1 1 trivial
847.2.f.v 16 11.c even 5 1 inner
847.2.f.w 16 11.b odd 2 1
847.2.f.w 16 11.d odd 10 1
847.2.f.x 16 11.c even 5 2
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 $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}^{16} + 2 T_{2}^{15} + 4 T_{2}^{14} + 3 T_{2}^{13} + 31 T_{2}^{12} + 70 T_{2}^{11} + 265 T_{2}^{10} + 385 T_{2}^{9} + 840 T_{2}^{8} + 975 T_{2}^{7} + 1085 T_{2}^{6} + 200 T_{2}^{5} + 275 T_{2}^{4} - 325 T_{2}^{3} + 450 T_{2}^{2} + 50 T_{2} + 25$$ T2^16 + 2*T2^15 + 4*T2^14 + 3*T2^13 + 31*T2^12 + 70*T2^11 + 265*T2^10 + 385*T2^9 + 840*T2^8 + 975*T2^7 + 1085*T2^6 + 200*T2^5 + 275*T2^4 - 325*T2^3 + 450*T2^2 + 50*T2 + 25 $$T_{3}^{16} + 2 T_{3}^{15} + 9 T_{3}^{14} + 26 T_{3}^{13} + 104 T_{3}^{12} + 120 T_{3}^{11} + 682 T_{3}^{10} + 1762 T_{3}^{9} + 6213 T_{3}^{8} + 10062 T_{3}^{7} + 18112 T_{3}^{6} + 11160 T_{3}^{5} + 32304 T_{3}^{4} - 2464 T_{3}^{3} + \cdots + 256$$ T3^16 + 2*T3^15 + 9*T3^14 + 26*T3^13 + 104*T3^12 + 120*T3^11 + 682*T3^10 + 1762*T3^9 + 6213*T3^8 + 10062*T3^7 + 18112*T3^6 + 11160*T3^5 + 32304*T3^4 - 2464*T3^3 + 3584*T3^2 - 1408*T3 + 256 $$T_{13}^{16} + 13 T_{13}^{15} + 124 T_{13}^{14} + 829 T_{13}^{13} + 4689 T_{13}^{12} + 21415 T_{13}^{11} + 84482 T_{13}^{10} + 254273 T_{13}^{9} + 648843 T_{13}^{8} + 1416018 T_{13}^{7} + 3693312 T_{13}^{6} + \cdots + 188897536$$ T13^16 + 13*T13^15 + 124*T13^14 + 829*T13^13 + 4689*T13^12 + 21415*T13^11 + 84482*T13^10 + 254273*T13^9 + 648843*T13^8 + 1416018*T13^7 + 3693312*T13^6 + 10037080*T13^5 + 27560624*T13^4 + 52221664*T13^3 + 99691904*T13^2 + 159320448*T13 + 188897536

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 2 T^{15} + 4 T^{14} + 3 T^{13} + \cdots + 25$$
$3$ $$T^{16} + 2 T^{15} + 9 T^{14} + 26 T^{13} + \cdots + 256$$
$5$ $$T^{16} + 5 T^{15} + 34 T^{14} + 169 T^{13} + \cdots + 256$$
$7$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{4}$$
$11$ $$T^{16}$$
$13$ $$T^{16} + 13 T^{15} + \cdots + 188897536$$
$17$ $$T^{16} + 10 T^{15} + 31 T^{14} + \cdots + 2611456$$
$19$ $$T^{16} - 6 T^{15} + 79 T^{14} + \cdots + 62726400$$
$23$ $$(T^{8} - 16 T^{7} + 40 T^{6} + 342 T^{5} + \cdots + 859)^{2}$$
$29$ $$T^{16} - 12 T^{15} + 46 T^{14} + \cdots + 245025$$
$31$ $$T^{16} + 2 T^{15} + \cdots + 2629638400$$
$37$ $$T^{16} + 11 T^{15} + 74 T^{14} + \cdots + 212521$$
$41$ $$T^{16} + 20 T^{15} + 171 T^{14} + \cdots + 13424896$$
$43$ $$(T^{8} - 4 T^{7} - 105 T^{6} + 268 T^{5} + \cdots - 971)^{2}$$
$47$ $$T^{16} - 7 T^{15} + \cdots + 5345713926400$$
$53$ $$T^{16} + 41 T^{15} + \cdots + 310840815961$$
$59$ $$T^{16} + 18 T^{15} + \cdots + 187142400$$
$61$ $$T^{16} - 12 T^{15} + \cdots + 253446400$$
$67$ $$(T^{8} + 19 T^{7} - 16 T^{6} - 1160 T^{5} + \cdots - 27395)^{2}$$
$71$ $$T^{16} - T^{15} - 71 T^{14} + \cdots + 379119841$$
$73$ $$T^{16} + 60 T^{15} + \cdots + 105069332736$$
$79$ $$T^{16} - 15 T^{15} + \cdots + 15858514175625$$
$83$ $$T^{16} + 20 T^{15} + \cdots + 756470016$$
$89$ $$(T^{8} - 37 T^{7} + 320 T^{6} + \cdots + 952400)^{2}$$
$97$ $$T^{16} + 35 T^{15} + \cdots + 4647025244416$$