Properties

 Label 418.2.f.g Level $418$ Weight $2$ Character orbit 418.f Analytic conductor $3.338$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.f (of order $$5$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - x^{15} + 7 x^{14} - 13 x^{13} + 51 x^{12} - 74 x^{11} + 332 x^{10} - 614 x^{9} + 1832 x^{8} - 2960 x^{7} + 5348 x^{6} - 6872 x^{5} + 8232 x^{4} - 6344 x^{3} + 3984 x^{2} + \cdots + 16$$ x^16 - x^15 + 7*x^14 - 13*x^13 + 51*x^12 - 74*x^11 + 332*x^10 - 614*x^9 + 1832*x^8 - 2960*x^7 + 5348*x^6 - 6872*x^5 + 8232*x^4 - 6344*x^3 + 3984*x^2 - 400*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{10} - \beta_{8} + \beta_{6} + 1) q^{2} + \beta_1 q^{3} + \beta_{6} q^{4} + ( - \beta_{15} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{5} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{6} + ( - \beta_{15} - \beta_{13} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{4} - 2) q^{7} + \beta_{8} q^{8} + (\beta_{13} - \beta_{12} - \beta_{11} + \beta_{6} - \beta_{5} + \beta_{4}) q^{9}+O(q^{10})$$ q + (b10 - b8 + b6 + 1) * q^2 + b1 * q^3 + b6 * q^4 + (-b15 + b12 + b11 + b9 - b6 - b4 - b3 - b2 - b1) * q^5 + (b4 + b3 + b2 + b1) * q^6 + (-b15 - b13 - b10 - b7 + b6 + b4 - 2) * q^7 + b8 * q^8 + (b13 - b12 - b11 + b6 - b5 + b4) * q^9 $$q + (\beta_{10} - \beta_{8} + \beta_{6} + 1) q^{2} + \beta_1 q^{3} + \beta_{6} q^{4} + ( - \beta_{15} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{5} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{6} + ( - \beta_{15} - \beta_{13} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{4} - 2) q^{7} + \beta_{8} q^{8} + (\beta_{13} - \beta_{12} - \beta_{11} + \beta_{6} - \beta_{5} + \beta_{4}) q^{9} + ( - \beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} + \beta_{9} + \beta_{7} - \beta_{6} - \beta_{3} - \beta_1) q^{10} + ( - 2 \beta_{9} - \beta_{7} + \beta_1 - 1) q^{11} + \beta_{3} q^{12} + ( - 2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{9} + 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \cdots - \beta_1) q^{13}+ \cdots + (\beta_{15} + \beta_{12} + \beta_{11} - 2 \beta_{10} + \beta_{9} + 4 \beta_{8} - \beta_{7} - 4 \beta_{6} + \cdots - 2 \beta_1) q^{99}+O(q^{100})$$ q + (b10 - b8 + b6 + 1) * q^2 + b1 * q^3 + b6 * q^4 + (-b15 + b12 + b11 + b9 - b6 - b4 - b3 - b2 - b1) * q^5 + (b4 + b3 + b2 + b1) * q^6 + (-b15 - b13 - b10 - b7 + b6 + b4 - 2) * q^7 + b8 * q^8 + (b13 - b12 - b11 + b6 - b5 + b4) * q^9 + (-b15 + b14 + b12 + b11 + b9 + b7 - b6 - b3 - b1) * q^10 + (-2*b9 - b7 + b1 - 1) * q^11 + b3 * q^12 + (-2*b15 + b14 + b13 + b9 + 2*b7 - b6 + b5 - b3 + b2 - b1) * q^13 + (b14 + b13 - b11 - b10 + 2*b8 - b5 + b4 - b3 - 1) * q^14 + (-2*b15 + b14 - b13 + b12 - b11 - 2*b10 - b6 + b5 - b3 + b2 + b1 - 3) * q^15 + b10 * q^16 + (-b14 - b13 - b12 + 2*b10 + 2*b9 + b7 - b5 - 2*b4 - b2 - 3*b1 + 2) * q^17 + (b15 + b13 - b9 - b5 - b2) * q^18 - b8 * q^19 + (-b15 + b14 - b13 + b12 + b11 + b9 + b7 - b6 + b5 + b2 - b1) * q^20 + (-b14 - b13 - b12 - b9 - b7 - b4 + 3*b3 - b1 + 1) * q^21 + (-b13 - 2*b7 + b1 - 2) * q^22 + (-b12 - b9 + b8 - b6 - b5 + 3*b4 + b3 + 2*b1 - 3) * q^23 - b2 * q^24 + (-b15 + b12 - 4*b10 - b9 + b8 - b7 - 4*b6 + b5 - b3 - b2 - 1) * q^25 + (2*b14 - b13 - b12 + b11 - b10 + b7 - 2*b6 - b5 + b4 + b3 + b2 - 2*b1 - 2) * q^26 + (-b13 + b11 + b10 + b9 - b8 - b6 - b5 + b4 + 2*b3 + b2 + 1) * q^27 + (b13 + b12 + 2*b10 - b9 - b8 + b5 - b4 - b3 - b2 + b1 + 1) * q^28 + (b15 - b14 + b13 + b10 - b9 - b7 + 2*b6 - b5 - b4 - 3*b2 - b1 + 1) * q^29 + (-b15 + 2*b14 + b13 - 2*b11 - 2*b10 - b9 + 2*b4 - b3 + b2 + 3*b1 - 2) * q^30 + (b15 - b14 + 2*b13 - 2*b12 - 2*b11 - b9 - b7 + 3*b6 - 2*b5 + b4 + b1 + 1) * q^31 - q^32 + (2*b15 - 3*b14 + 3*b10 - 2*b9 - b7 + 2*b6 + b5 + 2*b3 - b2 + 3*b1 + 1) * q^33 + (b15 + b13 - b11 + 2*b7 + b6 - 2*b5 + b4 - b3) * q^34 + (-b13 + b12 + 2*b11 + b10 - b8 - b6 + 2*b5 - b4 + b3 + 2*b2 + 1) * q^35 + (-b14 - b13 + b12 + b11 - b7 - b6 + b5 - b4 - b2) * q^36 + (-b13 - b12 + b11 + b10 + b9 + 2*b6 + b3 + 2*b2 + 1) * q^37 - b10 * q^38 + (b14 + b13 + b12 + 2*b10 - 2*b9 - 3*b8 - b7 + b5 - 2*b4 - 2*b3 - 3*b2 + b1 + 1) * q^39 + (-b15 + b14 - b13 + b9 + b7 + b5 + b2 - b1) * q^40 + (2*b14 + b13 - 3*b12 - 2*b11 + b10 - 2*b8 + 3*b7 + 3*b6 - b5 + 2*b4 + 2*b2 - b1 + 1) * q^41 + (b15 - b12 - b11 + b10 - b8 - b7 + 2*b6 - 2*b5 - b3 - 3*b2) * q^42 + (2*b15 + b14 + 3*b13 - 2*b11 - b8 + b7 + 3*b6 - b5 - b4 - b3 + 6) * q^43 + (2*b13 - b11 + b6 - b3 + 2*b1) * q^44 + (-b12 - b9 - b7 + b4 + 2*b3 + b1 - 3) * q^45 + (b15 - b13 - 2*b10 + 2*b8 - b7 - 3*b6 - b5 + 3*b4 + 3*b3 + 2*b2 - 4) * q^46 + (2*b15 - b14 + b13 - 3*b12 + b11 - b10 + 2*b9 + 5*b8 + 3*b7 - 2*b6 - 3*b5 - b4 + b3 - 5*b1 + 4) * q^47 + b4 * q^48 + (b14 + 2*b13 + 4*b12 + 2*b11 + 2*b10 - 3*b9 - 4*b8 - b7 - 2*b6 + 2*b5 - 2*b4 - 3*b3 - 3*b2 + 2*b1 + 2) * q^49 + (-b15 + b14 + b10 - 4*b8 - b7 - 2*b3 - b2 - b1 + 2) * q^50 + (-4*b15 + 2*b14 - 5*b13 - b12 + b11 - 3*b10 + 4*b9 + b7 + 3*b5 + b4 + b3 + 4*b2 - 3*b1 - 4) * q^51 + (b15 - 2*b13 - b10 + b9 - b6 + b5 + b3 + b2) * q^52 + (2*b15 + b14 - 3*b12 - 3*b10 + b9 + 3*b8 - 2*b7 + b6 - 3*b5 + 3*b4 + 2*b3 + 4*b2 - b1 - 2) * q^53 + (b12 + b9 - b8 + b7 + b6 + b3 + 1) * q^54 + (b15 - b13 + b12 + b11 + b10 + 2*b9 + 5*b8 + 3*b7 - 3*b6 - b5 - 2*b4 - b3 + 2*b2 - 2*b1 - 1) * q^55 + (-b15 - b13 + b11 - b8 - b7 + b5 - 2) * q^56 + b2 * q^57 + (-b14 + b12 + b11 + b10 + b8 - b7 - b4 - 2*b3 - 3*b2 - b1) * q^58 + (b15 - b14 - b12 + b11 + 2*b9 + b7 + b5 - 3*b4 + b3 + b2 - b1 + 2) * q^59 + (b14 + b13 - b11 - 2*b9 - 2*b8 - b7 + b6 + b5 + 2*b4 + b3 + b2 + 4*b1) * q^60 + (-b15 - b13 + b11 + 2*b10 + 2*b9 + 4*b8 - b6 - b5 - 2*b3 - 4*b1 - 4) * q^61 + (2*b15 - b14 + 2*b13 + b10 - 2*b9 - b7 + b6 - 2*b5 - b4 - 2*b2 + b1 + 1) * q^62 + (-2*b14 - 3*b13 + 3*b12 + 2*b11 + 2*b10 - 3*b8 - 3*b7 + 3*b5 - 2*b4 - 2*b2 + b1 - 1) * q^63 + (-b10 + b8 - b6 - 1) * q^64 + (2*b15 + 2*b13 - 3*b12 - 2*b11 - 3*b9 - 5*b8 - b7 + 7*b6 - 2*b5 + 2*b4 + b3 + 2*b1 + 5) * q^65 + (-2*b14 - b13 - b12 - b10 + 3*b8 - 2*b7 - b6 - b5 + b4 + 2*b3 - b1 - 4) * q^66 + (4*b15 - 2*b14 + 2*b13 + b12 - 4*b11 + b9 + 3*b8 + b7 + b6 + b4 + 5*b1 - 1) * q^67 + (-b14 - b13 + 2*b12 - 2*b10 - b9 + 2*b8 - 2*b6 + b5 + b3 + b2 + b1 - 1) * q^68 + (b14 + 3*b13 - 2*b12 - b11 - 6*b10 + 5*b8 + 2*b7 - 5*b6 - 3*b5 + b4 - 3*b3 - 2*b2 - 5*b1 + 1) * q^69 + (-b15 - 2*b13 - b12 + b11 + 2*b9 + b5 - 2*b4 + b3 + b2 - 2*b1) * q^70 + (-b15 - 3*b14 + b13 + 2*b12 + b11 + b10 + 3*b9 + 2*b8 + 3*b7 - b6 + b5 - 6*b4 - 3*b3 - 3*b2 - 4*b1 + 4) * q^71 + (-b15 + b9 - b3 - b1) * q^72 + (b15 + 3*b13 + 2*b12 - 2*b11 + 3*b7 + 3*b6 + 2*b5 - 2*b3 - b2 + b1 + 3) * q^73 + (b15 - b12 + b10 + b9 + 2*b8 + b7 + b6 - b5 + 2*b3 + 2*b2 + 1) * q^74 + (b15 - 2*b14 + b12 + 3*b10 - 2*b9 - 3*b8 - b7 + 4*b6 + b5 - 5*b4 - 3*b3 - 2*b2 + 2*b1 + 5) * q^75 + q^76 + (-3*b15 + 3*b14 + 2*b13 - b12 - b11 - 2*b10 + 2*b9 + 3*b8 + b7 - 2*b6 + 4*b4 - 4*b3 + 2*b2 + b1 - 4) * q^77 + (-b15 - b13 + b11 - 3*b8 - 2*b7 + 2*b6 + 2*b5 + b4 - b3 + 2*b1 - 1) * q^78 + (-b15 + 3*b14 - 4*b13 + 2*b12 + b11 - 4*b10 + 3*b9 + 4*b8 + b7 - 5*b6 - b5 - 2*b3 + 2*b2 - 3*b1 - 6) * q^79 + (b14 - b12 - b11 + b7 + b6 + b4 + b2) * q^80 + (2*b15 + b14 + 2*b13 - 3*b10 - 3*b9 - 6*b6 - 3*b5 + 2*b4 - b2 + b1 - 4) * q^81 + (3*b15 - b13 - 2*b12 - b11 - 2*b10 - 2*b9 + b8 + b6 - b5 + 2*b4 + 3*b3 + 2*b2 + b1 - 1) * q^82 + (-2*b15 + 3*b10 + 2*b9 - b8 + b4 - 3*b3 + b2 - 3*b1 + 1) * q^83 + (b15 - b14 + 2*b13 + b12 - b11 - b9 + 2*b6 + 2*b4 - b3 + 3*b1 + 1) * q^84 + (-2*b15 - b13 + b12 + b10 - 2*b9 - 5*b8 - b7 + b6 + 3*b5 - 2*b3 - 2*b2 - 3*b1 - 1) * q^85 + (-2*b14 + b13 + b12 + b11 + 5*b10 - 2*b9 - 5*b8 + 5*b6 + 3*b5 - 4*b4 - b3 - 3*b2 + 2*b1 + 8) * q^86 + (-2*b15 - b14 - 3*b13 + 3*b12 + 2*b11 + 3*b9 - 4*b8 + b7 + 2*b6 + 2*b5 - 2*b4 - b3 - 2*b1 + 3) * q^87 + (b13 + b11 - b9 - b6 + b3 + b1) * q^88 + (-4*b14 - 4*b13 - b12 - b9 - b8 - 5*b7 + b6 + 4*b5 - 2*b4 + 3*b3 + 2*b1 - 1) * q^89 + (b15 - b12 - 2*b10 + 2*b8 - b7 - 2*b6 - b5 + b4 + b3 - b2 - 3) * q^90 + (3*b15 - 3*b14 - 6*b13 + 3*b11 + 3*b9 - b8 - 3*b6 + 3*b5 - 3*b4 + 4*b3 + b2 + b1 + 3) * q^91 + (-b14 + b13 + b12 - b11 - b10 - 2*b6 + b5 - 3*b4 - b3 - 3*b2 - b1) * q^92 + (b15 - b14 - 2*b13 + 2*b11 + 4*b10 + 2*b9 + b8 + b7 - 2*b6 - 2*b5 - 4*b4 - 3*b2 - 5*b1 + 1) * q^93 + (3*b15 - 2*b14 - 2*b13 + 2*b11 + 5*b10 + b9 - b8 + 2*b7 - 2*b6 - 2*b5 - 2*b4 + 2*b3 - 4*b1 + 5) * q^94 + (b15 - b14 + b13 - b9 - b7 - b5 - b2 + b1) * q^95 - b1 * q^96 + (-2*b15 - b14 + 3*b12 + b11 - 3*b10 - b9 + 3*b8 + 2*b7 - 5*b6 + 4*b5 - 2*b4 + 2*b2 + b1 - 1) * q^97 + (-4*b15 - 4*b13 + 2*b12 + 4*b11 + 2*b9 - 4*b8 - 3*b7 + 5*b5 - 2*b4 - b3 - b1 - 1) * q^98 + (b15 + b12 + b11 - 2*b10 + b9 + 4*b8 - b7 - 4*b6 - b5 + b4 + b2 - 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 4 q^{2} + q^{3} - 4 q^{4} + q^{5} - q^{6} - 12 q^{7} + 4 q^{8} - q^{9}+O(q^{10})$$ 16 * q + 4 * q^2 + q^3 - 4 * q^4 + q^5 - q^6 - 12 * q^7 + 4 * q^8 - q^9 $$16 q + 4 q^{2} + q^{3} - 4 q^{4} + q^{5} - q^{6} - 12 q^{7} + 4 q^{8} - q^{9} + 4 q^{10} + 4 q^{11} - 4 q^{12} - 12 q^{13} + 12 q^{14} - 10 q^{15} - 4 q^{16} + 6 q^{17} + q^{18} - 4 q^{19} + q^{20} + 14 q^{21} - 9 q^{22} - 26 q^{23} - q^{24} + 33 q^{25} - 3 q^{26} + 13 q^{27} - 2 q^{28} + 10 q^{30} + 11 q^{31} - 16 q^{32} - 31 q^{33} - 6 q^{34} - 4 q^{35} - q^{36} + 4 q^{38} + 12 q^{39} - q^{40} - 7 q^{41} + 6 q^{42} + 66 q^{43} - 11 q^{44} - 42 q^{45} - 24 q^{46} + 47 q^{47} + q^{48} + 24 q^{49} + 37 q^{50} - 41 q^{51} + 3 q^{52} + 15 q^{53} - 8 q^{54} - 9 q^{55} - 28 q^{56} + q^{57} - 18 q^{59} + 5 q^{60} - 43 q^{61} + 9 q^{62} - 37 q^{63} - 4 q^{64} + 52 q^{65} - 39 q^{66} - 50 q^{67} + 6 q^{68} + 81 q^{69} - 11 q^{70} - 5 q^{71} + q^{72} - 13 q^{73} + 39 q^{75} + 16 q^{76} - 10 q^{77} - 22 q^{78} + 5 q^{79} + q^{80} + 2 q^{82} + 7 q^{83} - q^{84} - 29 q^{85} + 29 q^{86} + 4 q^{87} - 4 q^{88} - 28 q^{89} - 18 q^{90} - 11 q^{92} - 5 q^{93} + 28 q^{94} + q^{95} - q^{96} - 11 q^{97} - 24 q^{98} + 43 q^{99}+O(q^{100})$$ 16 * q + 4 * q^2 + q^3 - 4 * q^4 + q^5 - q^6 - 12 * q^7 + 4 * q^8 - q^9 + 4 * q^10 + 4 * q^11 - 4 * q^12 - 12 * q^13 + 12 * q^14 - 10 * q^15 - 4 * q^16 + 6 * q^17 + q^18 - 4 * q^19 + q^20 + 14 * q^21 - 9 * q^22 - 26 * q^23 - q^24 + 33 * q^25 - 3 * q^26 + 13 * q^27 - 2 * q^28 + 10 * q^30 + 11 * q^31 - 16 * q^32 - 31 * q^33 - 6 * q^34 - 4 * q^35 - q^36 + 4 * q^38 + 12 * q^39 - q^40 - 7 * q^41 + 6 * q^42 + 66 * q^43 - 11 * q^44 - 42 * q^45 - 24 * q^46 + 47 * q^47 + q^48 + 24 * q^49 + 37 * q^50 - 41 * q^51 + 3 * q^52 + 15 * q^53 - 8 * q^54 - 9 * q^55 - 28 * q^56 + q^57 - 18 * q^59 + 5 * q^60 - 43 * q^61 + 9 * q^62 - 37 * q^63 - 4 * q^64 + 52 * q^65 - 39 * q^66 - 50 * q^67 + 6 * q^68 + 81 * q^69 - 11 * q^70 - 5 * q^71 + q^72 - 13 * q^73 + 39 * q^75 + 16 * q^76 - 10 * q^77 - 22 * q^78 + 5 * q^79 + q^80 + 2 * q^82 + 7 * q^83 - q^84 - 29 * q^85 + 29 * q^86 + 4 * q^87 - 4 * q^88 - 28 * q^89 - 18 * q^90 - 11 * q^92 - 5 * q^93 + 28 * q^94 + q^95 - q^96 - 11 * q^97 - 24 * q^98 + 43 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{15} + 7 x^{14} - 13 x^{13} + 51 x^{12} - 74 x^{11} + 332 x^{10} - 614 x^{9} + 1832 x^{8} - 2960 x^{7} + 5348 x^{6} - 6872 x^{5} + 8232 x^{4} - 6344 x^{3} + 3984 x^{2} + \cdots + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 54\!\cdots\!43 \nu^{15} + \cdots + 19\!\cdots\!04 ) / 24\!\cdots\!20$$ (54113669090659143*v^15 + 125509485760498380*v^14 + 134170381609511754*v^13 + 408123460049112903*v^12 + 79310291113135720*v^11 + 4387249826154291929*v^10 + 3307207361107988776*v^9 + 21119795867192230058*v^8 - 19578954910849855143*v^7 + 141359824190214788062*v^6 - 266406687993577682136*v^5 + 412800346429034069584*v^4 - 649404215636808412292*v^3 + 631106189750884711240*v^2 - 463429991433631089112*v + 19715374402367819904) / 245730763696653461420 $$\beta_{3}$$ $$=$$ $$( - 16\!\cdots\!51 \nu^{15} + \cdots - 32\!\cdots\!04 ) / 49\!\cdots\!40$$ (-163107913533947251*v^15 + 212097294130585634*v^14 - 688747449401825354*v^13 + 2463901758595423776*v^12 - 6138599281361512122*v^11 + 11379816831792423305*v^10 - 39759283949379773780*v^9 + 103417297404752133378*v^8 - 205440624458721629604*v^7 + 423335269776883072400*v^6 - 422499347601014124560*v^5 + 788201989241779970456*v^4 - 634628949268406003024*v^3 + 333443578556107867048*v^2 - 32706475453281566208*v - 32755670404769134304) / 491461527393306922840 $$\beta_{4}$$ $$=$$ $$( - 22\!\cdots\!11 \nu^{15} + \cdots + 54\!\cdots\!20 ) / 49\!\cdots\!40$$ (-226050013263220611*v^15 - 115917867894302873*v^14 - 1450532074515444067*v^13 + 633196335154326083*v^12 - 8579531247879560741*v^11 + 1640636835235520554*v^10 - 59295754672617197520*v^9 + 37957614978304177644*v^8 - 273590101400418215272*v^7 + 147915974313033770020*v^6 - 552803460023140540088*v^5 + 272402976341914570188*v^4 - 566367517618225615592*v^3 - 5024814266148879896*v^2 - 486591638877690798784*v + 54763297462023210720) / 491461527393306922840 $$\beta_{5}$$ $$=$$ $$( - 26\!\cdots\!92 \nu^{15} + \cdots + 83\!\cdots\!12 ) / 49\!\cdots\!40$$ (-264507339439672892*v^15 + 833328660102107599*v^14 - 2795815056659971062*v^13 + 6712404472715380514*v^12 - 22186181832188849210*v^11 + 48255195114086564924*v^10 - 133532803015898157427*v^9 + 335929547570774966878*v^8 - 889524154216800589564*v^7 + 1772616906661178625212*v^6 - 3130865625249022845920*v^5 + 4530881455646203239432*v^4 - 5416191311383104005712*v^3 + 5562364548215835687184*v^2 - 3535209633711805859568*v + 830949447046808265312) / 491461527393306922840 $$\beta_{6}$$ $$=$$ $$( 20\!\cdots\!94 \nu^{15} + \cdots - 85\!\cdots\!08 ) / 49\!\cdots\!40$$ (2047229400298070894*v^15 - 2210337313832018145*v^14 + 14542703096217081892*v^13 - 27302729653276746976*v^12 + 106872601173797039370*v^11 - 157633574903418758278*v^10 + 691059977730751960113*v^9 - 1296758135732395302696*v^8 + 3853941558750818011186*v^7 - 6265239649341011475844*v^6 + 11371918102570966213512*v^5 - 14491059786449357308128*v^4 + 17640994412495499569864*v^3 - 13622252264759367754560*v^2 + 8489605509343622308744*v - 851598235572509923808) / 491461527393306922840 $$\beta_{7}$$ $$=$$ $$( - 20\!\cdots\!39 \nu^{15} + \cdots - 93\!\cdots\!80 ) / 49\!\cdots\!40$$ (-2048364697453358039*v^15 + 1604207159279922234*v^14 - 12995908732847054502*v^13 + 23096577785319715854*v^12 - 94109015645622297760*v^11 + 121321472235378343585*v^10 - 612910624300598625872*v^9 + 1080402599906383516598*v^8 - 3258242286224006109454*v^7 + 4898875541808936289042*v^6 - 8570778239995868692772*v^5 + 10425670191174525689872*v^4 - 11678951470821583658472*v^3 + 7522540943049970923544*v^2 - 3216776506284765418608*v - 937386809390689999080) / 491461527393306922840 $$\beta_{8}$$ $$=$$ $$( 12\!\cdots\!44 \nu^{15} + \cdots - 29\!\cdots\!88 ) / 24\!\cdots\!20$$ (1232210900147988744*v^15 - 1286324569238647887*v^14 + 8499966815275422828*v^13 - 16152912083533365426*v^12 + 62434632447498313041*v^11 - 91262916902064302776*v^10 + 404706769022977971079*v^9 - 759884700051973077592*v^8 + 2236290573203923148950*v^7 - 3627765309527196827097*v^6 + 6448504069801229014850*v^5 - 8201346617823400966632*v^4 + 9730759783589209271024*v^3 - 7167741734902032179644*v^2 + 4278022036438702444856*v - 29454368625564408488) / 245730763696653461420 $$\beta_{9}$$ $$=$$ $$( - 26\!\cdots\!97 \nu^{15} + \cdots + 16\!\cdots\!00 ) / 49\!\cdots\!40$$ (-2682608289538204997*v^15 + 3552704279509488571*v^14 - 19344844884499727648*v^13 + 40266728755699545302*v^12 - 146138652715870438812*v^11 + 236912324089506326961*v^10 - 937811999233356963363*v^9 + 1898499506893804610286*v^8 - 5349523097718146091510*v^7 + 9251589713682525429862*v^6 - 16283562053195642841152*v^5 + 21783528070146085568624*v^4 - 26267183198149791736976*v^3 + 21228099254726853237400*v^2 - 13303950575720602172488*v + 1665893353806043657200) / 491461527393306922840 $$\beta_{10}$$ $$=$$ $$( - 34\!\cdots\!70 \nu^{15} + \cdots + 88\!\cdots\!16 ) / 49\!\cdots\!40$$ (-3422706091376450670*v^15 + 3196656078113230059*v^14 - 24074860507529457563*v^13 + 43044647113378414643*v^12 - 173924814325044658087*v^11 + 244700719513977788839*v^10 - 1134697785501746101886*v^9 + 2042245785432523513860*v^8 - 6232439944423353449796*v^7 + 9857619929073875767928*v^6 - 18156716202368224413140*v^5 + 22968032799915828464152*v^4 - 27903313567869027345252*v^3 + 21147279926073977434888*v^2 - 13641085882309928349176*v + 882490797672889469216) / 491461527393306922840 $$\beta_{11}$$ $$=$$ $$( 43\!\cdots\!42 \nu^{15} + \cdots - 18\!\cdots\!72 ) / 49\!\cdots\!40$$ (4372116723935404242*v^15 - 4334896272913881631*v^14 + 29725931776654405218*v^13 - 56757472636933081514*v^12 + 217860167922777994176*v^11 - 316165968927527276772*v^10 + 1421847511719938479913*v^9 - 2654732075565875020664*v^8 + 7783184974393358840238*v^7 - 12649117363011595470980*v^6 + 22405996346437322690888*v^5 - 28893451368891717567636*v^4 + 34206900138813119809432*v^3 - 25895366143106099580088*v^2 + 16614750333162128376880*v - 1842926916357684716472) / 491461527393306922840 $$\beta_{12}$$ $$=$$ $$( 44\!\cdots\!39 \nu^{15} + \cdots - 59\!\cdots\!68 ) / 49\!\cdots\!40$$ (4494468988073864439*v^15 - 4661364094929838702*v^14 + 31983678495903535382*v^13 - 58420460766281394254*v^12 + 233283485073031644088*v^11 - 337104760994278011561*v^10 + 1507444401578495556680*v^9 - 2783813160327646298138*v^8 + 8389655168670458480386*v^7 - 13422829466135151487574*v^6 + 24535141683963264575784*v^5 - 30916930879548628639104*v^4 + 37178448953746113014000*v^3 - 28111598321244788445488*v^2 + 16454635281347167912240*v - 592117210457388024768) / 491461527393306922840 $$\beta_{13}$$ $$=$$ $$( - 23\!\cdots\!43 \nu^{15} + \cdots + 46\!\cdots\!16 ) / 24\!\cdots\!20$$ (-2369398244294163143*v^15 + 2420113591130115674*v^14 - 16887324465696666134*v^13 + 31173614634564260598*v^12 - 122550016450514711986*v^11 + 179878169252263711327*v^10 - 797579575113224567610*v^9 + 1475977990921783936086*v^8 - 4425137032372264137192*v^7 + 7180863224452091605658*v^6 - 13027130477797704970328*v^5 + 16789077382326559519248*v^4 - 20138492640678492044430*v^3 + 15627836041819896668876*v^2 - 9795705613599786935420*v + 468264369288334046416) / 245730763696653461420 $$\beta_{14}$$ $$=$$ $$( 60\!\cdots\!98 \nu^{15} + \cdots - 67\!\cdots\!88 ) / 49\!\cdots\!40$$ (6017578051910540698*v^15 - 5267190368980360139*v^14 + 40980541320911048494*v^13 - 73318481394592742340*v^12 + 295797203286858555740*v^11 - 405689949063254732678*v^10 + 1933940594396456552375*v^9 - 3452478213876622344434*v^8 + 10492359507346912926130*v^7 - 16408537803366494515206*v^6 + 29790661507938304194344*v^5 - 37390265319375055064320*v^4 + 44466030393119037302516*v^3 - 32468413285954840591624*v^2 + 19696076443869293752832*v - 671715934511648201488) / 491461527393306922840 $$\beta_{15}$$ $$=$$ $$( 81\!\cdots\!10 \nu^{15} + \cdots - 95\!\cdots\!20 ) / 49\!\cdots\!40$$ (8107864207827409310*v^15 - 6738611853010991734*v^14 + 55791686403945833919*v^13 - 96692088008052083549*v^12 + 398557712218351184593*v^11 - 537886712838201156261*v^10 + 2614731818598975003795*v^9 - 4569085540178392874910*v^8 + 14156309929407767590166*v^7 - 21856218523896830504122*v^6 + 40178424862350131923388*v^5 - 50098508198025408749224*v^4 + 60026193067649748930668*v^3 - 43905221800232821163960*v^2 + 27243273703346420472616*v - 950555006218756708720) / 491461527393306922840
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{13} - \beta_{12} - \beta_{11} - 3\beta_{10} + 3\beta_{8} - 2\beta_{6} - \beta_{5} + \beta_{4} - 3$$ b13 - b12 - b11 - 3*b10 + 3*b8 - 2*b6 - b5 + b4 - 3 $$\nu^{3}$$ $$=$$ $$- \beta_{13} + \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - 5 \beta_{4} - 4 \beta_{3} - 5 \beta_{2} - 6 \beta _1 + 1$$ -b13 + b11 + b10 + b9 - b8 - b6 - b5 - 5*b4 - 4*b3 - 5*b2 - 6*b1 + 1 $$\nu^{4}$$ $$=$$ $$- 7 \beta_{15} + \beta_{14} - 7 \beta_{13} - 3 \beta_{10} + 6 \beta_{9} + 12 \beta_{6} + 6 \beta_{5} + 2 \beta_{4} + 8 \beta_{2} + \beta _1 - 4$$ -7*b15 + b14 - 7*b13 - 3*b10 + 6*b9 + 12*b6 + 6*b5 + 2*b4 + 8*b2 + b1 - 4 $$\nu^{5}$$ $$=$$ $$2 \beta_{15} + 2 \beta_{13} - 9 \beta_{12} - 2 \beta_{11} - 9 \beta_{9} + 9 \beta_{8} - 7 \beta_{7} - 7 \beta_{6} - 2 \beta_{5} + 27 \beta_{3} - 7$$ 2*b15 + 2*b13 - 9*b12 - 2*b11 - 9*b9 + 9*b8 - 7*b7 - 7*b6 - 2*b5 + 27*b3 - 7 $$\nu^{6}$$ $$=$$ $$2 \beta_{15} - 43 \beta_{14} - 36 \beta_{13} + 34 \beta_{12} + 43 \beta_{11} + 27 \beta_{10} + 2 \beta_{9} - 83 \beta_{8} - 34 \beta_{7} - 16 \beta_{6} + 34 \beta_{5} - 43 \beta_{4} - 9 \beta_{3} - 52 \beta_{2} - 9 \beta _1 + 9$$ 2*b15 - 43*b14 - 36*b13 + 34*b12 + 43*b11 + 27*b10 + 2*b9 - 83*b8 - 34*b7 - 16*b6 + 34*b5 - 43*b4 - 9*b3 - 52*b2 - 9*b1 + 9 $$\nu^{7}$$ $$=$$ $$- 63 \beta_{15} + 22 \beta_{14} + 2 \beta_{13} + 39 \beta_{12} - 2 \beta_{11} - 69 \beta_{10} + 22 \beta_{9} + 69 \beta_{8} + 63 \beta_{7} + 39 \beta_{5} + 24 \beta_{4} + 175 \beta_{2} - 22 \beta _1 + 39$$ -63*b15 + 22*b14 + 2*b13 + 39*b12 - 2*b11 - 69*b10 + 22*b9 + 69*b8 + 63*b7 + 39*b5 + 24*b4 + 175*b2 - 22*b1 + 39 $$\nu^{8}$$ $$=$$ $$194 \beta_{15} + 22 \beta_{14} + 39 \beta_{13} - 39 \beta_{11} + 483 \beta_{10} - 255 \beta_{9} - 191 \beta_{8} - 22 \beta_{7} + 39 \beta_{6} + 39 \beta_{5} - 73 \beta_{4} + 190 \beta_{3} - 95 \beta_{2} + 290 \beta _1 + 147$$ 194*b15 + 22*b14 + 39*b13 - 39*b11 + 483*b10 - 255*b9 - 191*b8 - 22*b7 + 39*b6 + 39*b5 - 73*b4 + 190*b3 - 95*b2 + 290*b1 + 147 $$\nu^{9}$$ $$=$$ $$199 \beta_{15} - 407 \beta_{14} + 199 \beta_{13} - 467 \beta_{10} + 34 \beta_{9} - 174 \beta_{7} + 34 \beta_{6} + 34 \beta_{5} + 660 \beta_{4} + 34 \beta_{2} + 407 \beta _1 - 234$$ 199*b15 - 407*b14 + 199*b13 - 467*b10 + 34*b9 - 174*b7 + 34*b6 + 34*b5 + 660*b4 + 34*b2 + 407*b1 - 234 $$\nu^{10}$$ $$=$$ $$1126 \beta_{14} + 1126 \beta_{13} + 199 \beta_{12} + 199 \beta_{9} + 1255 \beta_{8} + 1499 \beta_{7} - 1255 \beta_{6} - 1300 \beta_{5} - 874 \beta_{4} - 1257 \beta_{3} - 2174 \beta _1 + 2005$$ 1126*b14 + 1126*b13 + 199*b12 + 199*b9 + 1255*b8 + 1499*b7 - 1255*b6 - 1300*b5 - 874*b4 - 1257*b3 - 2174*b1 + 2005 $$\nu^{11}$$ $$=$$ $$955 \beta_{14} - 1592 \beta_{13} + 370 \beta_{12} - 955 \beta_{11} + 3151 \beta_{10} - 3529 \beta_{8} - 370 \beta_{7} + 4106 \beta_{6} + 1592 \beta_{5} + 955 \beta_{4} - 1333 \beta_{3} - 378 \beta_{2} + 5305 \beta _1 - 1325$$ 955*b14 - 1592*b13 + 370*b12 - 955*b11 + 3151*b10 - 3529*b8 - 370*b7 + 4106*b6 + 1592*b5 + 955*b4 - 1333*b3 - 378*b2 + 5305*b1 - 1325 $$\nu^{12}$$ $$=$$ $$947 \beta_{15} + 8 \beta_{14} + 7860 \beta_{13} - 8815 \beta_{12} - 6638 \beta_{11} - 17531 \beta_{10} + 8 \beta_{9} + 17531 \beta_{8} - 947 \beta_{7} - 2858 \beta_{6} - 7593 \beta_{5} + 12336 \beta_{4} + 5690 \beta_{3} + \cdots - 10451$$ 947*b15 + 8*b14 + 7860*b13 - 8815*b12 - 6638*b11 - 17531*b10 + 8*b9 + 17531*b8 - 947*b7 - 2858*b6 - 7593*b5 + 12336*b4 + 5690*b3 + 5499*b2 - 8*b1 - 10451 $$\nu^{13}$$ $$=$$ $$3314 \beta_{15} + 8 \beta_{14} - 7593 \beta_{13} + 8134 \beta_{12} + 15727 \beta_{11} + 24341 \beta_{10} + 4271 \beta_{9} - 20867 \beta_{8} - 8 \beta_{7} - 15727 \beta_{6} - 7593 \beta_{5} - 40163 \beta_{4} + \cdots + 20851$$ 3314*b15 + 8*b14 - 7593*b13 + 8134*b12 + 15727*b11 + 24341*b10 + 4271*b9 - 20867*b8 - 8*b7 - 15727*b6 - 7593*b5 - 40163*b4 - 20954*b3 - 40171*b2 - 36132*b1 + 20851 $$\nu^{14}$$ $$=$$ $$- 52035 \beta_{15} + 4087 \beta_{14} - 43901 \beta_{13} + 8134 \beta_{12} - 8134 \beta_{11} - 51023 \beta_{10} + 39622 \beta_{9} - 192 \beta_{7} + 64678 \beta_{6} + 47756 \beta_{5} + 43390 \beta_{4} + \cdots - 55302$$ -52035*b15 + 4087*b14 - 43901*b13 + 8134*b12 - 8134*b11 - 51023*b10 + 39622*b9 - 192*b7 + 64678*b6 + 47756*b5 + 43390*b4 - 8134*b3 + 76032*b2 + 40457*b1 - 55302 $$\nu^{15}$$ $$=$$ $$52678 \beta_{15} + 26698 \beta_{14} + 79376 \beta_{13} - 96579 \beta_{12} - 52678 \beta_{11} - 96579 \beta_{9} + 136591 \beta_{8} - 17011 \beta_{7} - 83913 \beta_{6} - 79568 \beta_{5} - 2392 \beta_{4} + \cdots - 15019$$ 52678*b15 + 26698*b14 + 79376*b13 - 96579*b12 - 52678*b11 - 96579*b9 + 136591*b8 - 17011*b7 - 83913*b6 - 79568*b5 - 2392*b4 + 166047*b3 - 29282*b1 - 15019

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/418\mathbb{Z}\right)^\times$$.

 $$n$$ $$287$$ $$343$$ $$\chi(n)$$ $$1$$ $$\beta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
115.1
 −2.04714 − 1.48734i 0.0560397 + 0.0407152i 0.956027 + 0.694595i 1.84409 + 1.33981i −0.744204 + 2.29042i −0.494109 + 1.52071i 0.388450 − 1.19553i 0.540846 − 1.66455i −2.04714 + 1.48734i 0.0560397 − 0.0407152i 0.956027 − 0.694595i 1.84409 − 1.33981i −0.744204 − 2.29042i −0.494109 − 1.52071i 0.388450 + 1.19553i 0.540846 + 1.66455i
−0.309017 0.951057i −2.04714 1.48734i −0.809017 + 0.587785i 0.0815989 0.251136i −0.781939 + 2.40656i −3.52166 + 2.55863i 0.809017 + 0.587785i 1.05158 + 3.23642i −0.264060
115.2 −0.309017 0.951057i 0.0560397 + 0.0407152i −0.809017 + 0.587785i 0.340306 1.04736i 0.0214053 0.0658786i −1.61489 + 1.17328i 0.809017 + 0.587785i −0.925568 2.84861i −1.10125
115.3 −0.309017 0.951057i 0.956027 + 0.694595i −0.809017 + 0.587785i −0.0364140 + 0.112071i 0.365170 1.12388i 1.41067 1.02491i 0.809017 + 0.587785i −0.495524 1.52507i 0.117838
115.4 −0.309017 0.951057i 1.84409 + 1.33981i −0.809017 + 0.587785i −0.694508 + 2.13748i 0.704381 2.16786i −3.74627 + 2.72182i 0.809017 + 0.587785i 0.678534 + 2.08831i 2.24748
191.1 0.809017 + 0.587785i −0.744204 + 2.29042i 0.309017 + 0.951057i 2.77774 2.01814i −1.94835 + 1.41556i 0.326530 + 1.00496i −0.309017 + 0.951057i −2.26515 1.64573i 3.43347
191.2 0.809017 + 0.587785i −0.494109 + 1.52071i 0.309017 + 0.951057i −3.03200 + 2.20287i −1.29359 + 0.939851i −0.259070 0.797336i −0.309017 + 0.951057i 0.358634 + 0.260563i −3.74775
191.3 0.809017 + 0.587785i 0.388450 1.19553i 0.309017 + 0.951057i −1.36174 + 0.989363i 1.01697 0.738875i 1.33687 + 4.11446i −0.309017 + 0.951057i 1.14866 + 0.834553i −1.68320
191.4 0.809017 + 0.587785i 0.540846 1.66455i 0.309017 + 0.951057i 2.42502 1.76188i 1.41595 1.02875i 0.0678059 + 0.208685i −0.309017 + 0.951057i −0.0511674 0.0371753i 2.99749
229.1 −0.309017 + 0.951057i −2.04714 + 1.48734i −0.809017 0.587785i 0.0815989 + 0.251136i −0.781939 2.40656i −3.52166 2.55863i 0.809017 0.587785i 1.05158 3.23642i −0.264060
229.2 −0.309017 + 0.951057i 0.0560397 0.0407152i −0.809017 0.587785i 0.340306 + 1.04736i 0.0214053 + 0.0658786i −1.61489 1.17328i 0.809017 0.587785i −0.925568 + 2.84861i −1.10125
229.3 −0.309017 + 0.951057i 0.956027 0.694595i −0.809017 0.587785i −0.0364140 0.112071i 0.365170 + 1.12388i 1.41067 + 1.02491i 0.809017 0.587785i −0.495524 + 1.52507i 0.117838
229.4 −0.309017 + 0.951057i 1.84409 1.33981i −0.809017 0.587785i −0.694508 2.13748i 0.704381 + 2.16786i −3.74627 2.72182i 0.809017 0.587785i 0.678534 2.08831i 2.24748
267.1 0.809017 0.587785i −0.744204 2.29042i 0.309017 0.951057i 2.77774 + 2.01814i −1.94835 1.41556i 0.326530 1.00496i −0.309017 0.951057i −2.26515 + 1.64573i 3.43347
267.2 0.809017 0.587785i −0.494109 1.52071i 0.309017 0.951057i −3.03200 2.20287i −1.29359 0.939851i −0.259070 + 0.797336i −0.309017 0.951057i 0.358634 0.260563i −3.74775
267.3 0.809017 0.587785i 0.388450 + 1.19553i 0.309017 0.951057i −1.36174 0.989363i 1.01697 + 0.738875i 1.33687 4.11446i −0.309017 0.951057i 1.14866 0.834553i −1.68320
267.4 0.809017 0.587785i 0.540846 + 1.66455i 0.309017 0.951057i 2.42502 + 1.76188i 1.41595 + 1.02875i 0.0678059 0.208685i −0.309017 0.951057i −0.0511674 + 0.0371753i 2.99749
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 267.4 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 418.2.f.g 16
11.c even 5 1 inner 418.2.f.g 16
11.c even 5 1 4598.2.a.bw 8
11.d odd 10 1 4598.2.a.bz 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.g 16 1.a even 1 1 trivial
418.2.f.g 16 11.c even 5 1 inner
4598.2.a.bw 8 11.c even 5 1
4598.2.a.bz 8 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} - T_{3}^{15} + 7 T_{3}^{14} - 13 T_{3}^{13} + 51 T_{3}^{12} - 74 T_{3}^{11} + 332 T_{3}^{10} - 614 T_{3}^{9} + 1832 T_{3}^{8} - 2960 T_{3}^{7} + 5348 T_{3}^{6} - 6872 T_{3}^{5} + 8232 T_{3}^{4} - 6344 T_{3}^{3} + 3984 T_{3}^{2} + \cdots + 16$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{4}$$
$3$ $$T^{16} - T^{15} + 7 T^{14} - 13 T^{13} + \cdots + 16$$
$5$ $$T^{16} - T^{15} - 6 T^{14} + 3 T^{13} + \cdots + 25$$
$7$ $$T^{16} + 12 T^{15} + 74 T^{14} + \cdots + 3481$$
$11$ $$T^{16} - 4 T^{15} - 4 T^{14} + \cdots + 214358881$$
$13$ $$T^{16} + 12 T^{15} + 120 T^{14} + \cdots + 102400$$
$17$ $$T^{16} - 6 T^{15} + \cdots + 14059082041$$
$19$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{4}$$
$23$ $$(T^{8} + 13 T^{7} - 47 T^{6} + \cdots - 142921)^{2}$$
$29$ $$T^{16} + 74 T^{14} + \cdots + 1124663296$$
$31$ $$T^{16} - 11 T^{15} + \cdots + 528816016$$
$37$ $$T^{16} + 100 T^{14} + 367 T^{13} + \cdots + 6370576$$
$41$ $$T^{16} + 7 T^{15} + 106 T^{14} + \cdots + 146410000$$
$43$ $$(T^{8} - 33 T^{7} + 336 T^{6} + \cdots - 1634476)^{2}$$
$47$ $$T^{16} - 47 T^{15} + \cdots + 6705277197025$$
$53$ $$T^{16} - 15 T^{15} + \cdots + 10973819536$$
$59$ $$T^{16} + 18 T^{15} + 338 T^{14} + \cdots + 2560000$$
$61$ $$T^{16} + 43 T^{15} + \cdots + 49999196025$$
$67$ $$(T^{8} + 25 T^{7} - 165 T^{6} + \cdots - 72329500)^{2}$$
$71$ $$T^{16} + 5 T^{15} + \cdots + 5035015405456$$
$73$ $$T^{16} + 13 T^{15} + \cdots + 7776423813376$$
$79$ $$T^{16} - 5 T^{15} + \cdots + 7299831312400$$
$83$ $$T^{16} - 7 T^{15} + \cdots + 2600869023841$$
$89$ $$(T^{8} + 14 T^{7} - 233 T^{6} + \cdots - 481856)^{2}$$
$97$ $$T^{16} + 11 T^{15} + \cdots + 281430250000$$