# Properties

 Label 625.2.d.n Level $625$ Weight $2$ Character orbit 625.d Analytic conductor $4.991$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.d (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.99065012633$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 25x^{14} + 239x^{12} + 1165x^{10} + 3166x^{8} + 4820x^{6} + 3809x^{4} + 1205x^{2} + 1$$ x^16 + 25*x^14 + 239*x^12 + 1165*x^10 + 3166*x^8 + 4820*x^6 + 3809*x^4 + 1205*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$5^{2}$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 25x^{14} + 239x^{12} + 1165x^{10} + 3166x^{8} + 4820x^{6} + 3809x^{4} + 1205x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 29\nu^{14} + 619\nu^{12} + 4515\nu^{10} + 13949\nu^{8} + 15463\nu^{6} - 4230\nu^{4} - 12623\nu^{2} - 1474 ) / 801$$ (29*v^14 + 619*v^12 + 4515*v^10 + 13949*v^8 + 15463*v^6 - 4230*v^4 - 12623*v^2 - 1474) / 801 $$\beta_{2}$$ $$=$$ $$( -19\nu^{14} - 470\nu^{12} - 4422\nu^{10} - 20887\nu^{8} - 52142\nu^{6} - 63150\nu^{4} - 25961\nu^{2} + 1601 ) / 801$$ (-19*v^14 - 470*v^12 - 4422*v^10 - 20887*v^8 - 52142*v^6 - 63150*v^4 - 25961*v^2 + 1601) / 801 $$\beta_{3}$$ $$=$$ $$( 29 \nu^{15} - 29 \nu^{14} + 886 \nu^{13} - 619 \nu^{12} + 10389 \nu^{11} - 4515 \nu^{10} + 59873 \nu^{9} - 13949 \nu^{8} + 183139 \nu^{7} - 15463 \nu^{6} + 297480 \nu^{5} + 4230 \nu^{4} + \cdots + 673 ) / 1602$$ (29*v^15 - 29*v^14 + 886*v^13 - 619*v^12 + 10389*v^11 - 4515*v^10 + 59873*v^9 - 13949*v^8 + 183139*v^7 - 15463*v^6 + 297480*v^5 + 4230*v^4 + 237289*v^3 + 12623*v^2 + 70883*v + 673) / 1602 $$\beta_{4}$$ $$=$$ $$( 22 \nu^{15} + \nu^{14} + 488 \nu^{13} + 6 \nu^{12} + 3907 \nu^{11} - 142 \nu^{10} + 15210 \nu^{9} - 1388 \nu^{8} + 32176 \nu^{7} - 4282 \nu^{6} + 39643 \nu^{5} - 4869 \nu^{4} + 29676 \nu^{3} + \cdots + 431 ) / 534$$ (22*v^15 + v^14 + 488*v^13 + 6*v^12 + 3907*v^11 - 142*v^10 + 15210*v^9 - 1388*v^8 + 32176*v^7 - 4282*v^6 + 39643*v^5 - 4869*v^4 + 29676*v^3 - 1224*v^2 + 10906*v + 431) / 534 $$\beta_{5}$$ $$=$$ $$( - 22 \nu^{15} + \nu^{14} - 488 \nu^{13} + 6 \nu^{12} - 3907 \nu^{11} - 142 \nu^{10} - 15210 \nu^{9} - 1388 \nu^{8} - 32176 \nu^{7} - 4282 \nu^{6} - 39643 \nu^{5} - 4869 \nu^{4} - 29676 \nu^{3} + \cdots + 431 ) / 534$$ (-22*v^15 + v^14 - 488*v^13 + 6*v^12 - 3907*v^11 - 142*v^10 - 15210*v^9 - 1388*v^8 - 32176*v^7 - 4282*v^6 - 39643*v^5 - 4869*v^4 - 29676*v^3 - 1224*v^2 - 10906*v + 431) / 534 $$\beta_{6}$$ $$=$$ $$( 32 \nu^{15} - 16 \nu^{14} + 1082 \nu^{13} - 541 \nu^{12} + 13968 \nu^{11} - 6717 \nu^{10} + 88283 \nu^{9} - 38668 \nu^{8} + 297029 \nu^{7} - 110467 \nu^{6} + 534120 \nu^{5} + \cdots - 1556 ) / 1602$$ (32*v^15 - 16*v^14 + 1082*v^13 - 541*v^12 + 13968*v^11 - 6717*v^10 + 88283*v^9 - 38668*v^8 + 297029*v^7 - 110467*v^6 + 534120*v^5 - 150381*v^4 + 474718*v^3 - 77159*v^2 + 159574*v - 1556) / 1602 $$\beta_{7}$$ $$=$$ $$( 32 \nu^{15} + 16 \nu^{14} + 1082 \nu^{13} + 541 \nu^{12} + 13968 \nu^{11} + 6717 \nu^{10} + 88283 \nu^{9} + 38668 \nu^{8} + 297029 \nu^{7} + 110467 \nu^{6} + 534120 \nu^{5} + \cdots + 1556 ) / 1602$$ (32*v^15 + 16*v^14 + 1082*v^13 + 541*v^12 + 13968*v^11 + 6717*v^10 + 88283*v^9 + 38668*v^8 + 297029*v^7 + 110467*v^6 + 534120*v^5 + 150381*v^4 + 474718*v^3 + 77159*v^2 + 159574*v + 1556) / 1602 $$\beta_{8}$$ $$=$$ $$( - 23 \nu^{15} - 119 \nu^{14} - 316 \nu^{13} - 2583 \nu^{12} + 240 \nu^{11} - 19770 \nu^{10} + 18841 \nu^{9} - 70055 \nu^{8} + 100622 \nu^{7} - 120384 \nu^{6} + 227865 \nu^{5} + \cdots + 3891 ) / 1602$$ (-23*v^15 - 119*v^14 - 316*v^13 - 2583*v^12 + 240*v^11 - 19770*v^10 + 18841*v^9 - 70055*v^8 + 100622*v^7 - 120384*v^6 + 227865*v^5 - 90759*v^4 + 240239*v^3 - 17926*v^2 + 98578*v + 3891) / 1602 $$\beta_{9}$$ $$=$$ $$( 72 \nu^{15} + \nu^{14} + 1678 \nu^{13} + 6 \nu^{12} + 14429 \nu^{11} - 142 \nu^{10} + 60887 \nu^{9} - 1388 \nu^{8} + 136696 \nu^{7} - 4282 \nu^{6} + 161716 \nu^{5} - 4869 \nu^{4} + 91029 \nu^{3} + \cdots + 164 ) / 534$$ (72*v^15 + v^14 + 1678*v^13 + 6*v^12 + 14429*v^11 - 142*v^10 + 60887*v^9 - 1388*v^8 + 136696*v^7 - 4282*v^6 + 161716*v^5 - 4869*v^4 + 91029*v^3 - 1224*v^2 + 17682*v + 164) / 534 $$\beta_{10}$$ $$=$$ $$( 35 \nu^{15} - 250 \nu^{14} + 833 \nu^{13} - 5683 \nu^{12} + 7668 \nu^{11} - 46914 \nu^{10} + 38195 \nu^{9} - 185932 \nu^{8} + 116507 \nu^{7} - 377263 \nu^{6} + 217803 \nu^{5} + \cdots - 950 ) / 1602$$ (35*v^15 - 250*v^14 + 833*v^13 - 5683*v^12 + 7668*v^11 - 46914*v^10 + 38195*v^9 - 185932*v^8 + 116507*v^7 - 377263*v^6 + 217803*v^5 - 371667*v^4 + 221134*v^3 - 138110*v^2 + 89311*v - 950) / 1602 $$\beta_{11}$$ $$=$$ $$( - 52 \nu^{15} - 221 \nu^{14} - 1113 \nu^{13} - 5064 \nu^{12} - 8280 \nu^{11} - 42399 \nu^{10} - 27682 \nu^{9} - 171983 \nu^{8} - 41577 \nu^{7} - 361800 \nu^{6} - 19152 \nu^{5} + \cdots - 21 ) / 1602$$ (-52*v^15 - 221*v^14 - 1113*v^13 - 5064*v^12 - 8280*v^11 - 42399*v^10 - 27682*v^9 - 171983*v^8 - 41577*v^7 - 361800*v^6 - 19152*v^5 - 375897*v^4 + 11761*v^3 - 149932*v^2 + 11052*v - 21) / 1602 $$\beta_{12}$$ $$=$$ $$( 135 \nu^{14} + 3124 \nu^{12} + 26487 \nu^{10} + 108723 \nu^{8} + 230851 \nu^{6} + 241140 \nu^{4} + 95085 \nu^{2} - 2335 ) / 801$$ (135*v^14 + 3124*v^12 + 26487*v^10 + 108723*v^8 + 230851*v^6 + 241140*v^4 + 95085*v^2 - 2335) / 801 $$\beta_{13}$$ $$=$$ $$( 35 \nu^{15} + 250 \nu^{14} + 833 \nu^{13} + 5683 \nu^{12} + 7668 \nu^{11} + 46914 \nu^{10} + 38195 \nu^{9} + 185932 \nu^{8} + 116507 \nu^{7} + 377263 \nu^{6} + 217803 \nu^{5} + \cdots + 950 ) / 1602$$ (35*v^15 + 250*v^14 + 833*v^13 + 5683*v^12 + 7668*v^11 + 46914*v^10 + 38195*v^9 + 185932*v^8 + 116507*v^7 + 377263*v^6 + 217803*v^5 + 371667*v^4 + 221134*v^3 + 138110*v^2 + 89311*v + 950) / 1602 $$\beta_{14}$$ $$=$$ $$( - 14 \nu^{15} - 282 \nu^{14} - 351 \nu^{13} - 6320 \nu^{12} - 3441 \nu^{11} - 51003 \nu^{10} - 17948 \nu^{9} - 195717 \nu^{8} - 54951 \nu^{7} - 379880 \nu^{6} - 99777 \nu^{5} + \cdots + 1634 ) / 1602$$ (-14*v^15 - 282*v^14 - 351*v^13 - 6320*v^12 - 3441*v^11 - 51003*v^10 - 17948*v^9 - 195717*v^8 - 54951*v^7 - 379880*v^6 - 99777*v^5 - 352830*v^4 - 100789*v^3 - 121014*v^2 - 43770*v + 1634) / 1602 $$\beta_{15}$$ $$=$$ $$( - 187 \nu^{15} + 484 \nu^{14} - 4415 \nu^{13} + 10914 \nu^{12} - 38772 \nu^{11} + 88980 \nu^{10} - 168712 \nu^{9} + 346813 \nu^{8} - 394625 \nu^{7} + 689538 \nu^{6} + \cdots + 2391 ) / 1602$$ (-187*v^15 + 484*v^14 - 4415*v^13 + 10914*v^12 - 38772*v^11 + 88980*v^10 - 168712*v^9 + 346813*v^8 - 394625*v^7 + 689538*v^6 - 489378*v^5 + 665577*v^4 - 285710*v^3 + 245786*v^2 - 55321*v + 2391) / 1602
 $$\nu$$ $$=$$ $$( - 4 \beta_{15} - \beta_{14} + 3 \beta_{13} + \beta_{12} - 3 \beta_{11} - \beta_{10} - 4 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} - \beta_{6} + 4 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 3 \beta_1 ) / 5$$ (-4*b15 - b14 + 3*b13 + b12 - 3*b11 - b10 - 4*b9 + 2*b8 - 3*b7 - b6 + 4*b5 + 3*b4 + 4*b3 + 2*b2 + 3*b1) / 5 $$\nu^{2}$$ $$=$$ $$\beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} + \beta_{4} - 3$$ b14 + b13 + b11 - b10 + b4 - 3 $$\nu^{3}$$ $$=$$ $$( 26 \beta_{15} + \beta_{14} - 12 \beta_{13} - 4 \beta_{12} + 25 \beta_{11} + 14 \beta_{10} + 34 \beta_{9} - 8 \beta_{8} + 12 \beta_{7} + 4 \beta_{6} - 21 \beta_{5} - 38 \beta_{4} - 28 \beta_{3} - 13 \beta_{2} - 26 \beta _1 + 3 ) / 5$$ (26*b15 + b14 - 12*b13 - 4*b12 + 25*b11 + 14*b10 + 34*b9 - 8*b8 + 12*b7 + 4*b6 - 21*b5 - 38*b4 - 28*b3 - 13*b2 - 26*b1 + 3) / 5 $$\nu^{4}$$ $$=$$ $$- 11 \beta_{14} - 9 \beta_{13} - 2 \beta_{12} - 11 \beta_{11} + 9 \beta_{10} - \beta_{7} + \beta_{6} - 4 \beta_{5} - 15 \beta_{4} + 3 \beta_{2} - 5 \beta _1 + 18$$ -11*b14 - 9*b13 - 2*b12 - 11*b11 + 9*b10 - b7 + b6 - 4*b5 - 15*b4 + 3*b2 - 5*b1 + 18 $$\nu^{5}$$ $$=$$ $$( - 226 \beta_{15} + 6 \beta_{14} + 67 \beta_{13} + 34 \beta_{12} - 232 \beta_{11} - 159 \beta_{10} - 316 \beta_{9} + 68 \beta_{8} - 77 \beta_{7} - 9 \beta_{6} + 156 \beta_{5} + 392 \beta_{4} + 206 \beta_{3} + 113 \beta_{2} + 222 \beta _1 - 55 ) / 5$$ (-226*b15 + 6*b14 + 67*b13 + 34*b12 - 232*b11 - 159*b10 - 316*b9 + 68*b8 - 77*b7 - 9*b6 + 156*b5 + 392*b4 + 206*b3 + 113*b2 + 222*b1 - 55) / 5 $$\nu^{6}$$ $$=$$ $$109 \beta_{14} + 84 \beta_{13} + 21 \beta_{12} + 109 \beta_{11} - 84 \beta_{10} + 19 \beta_{7} - 19 \beta_{6} + 62 \beta_{5} + 171 \beta_{4} - 41 \beta_{2} + 74 \beta _1 - 155$$ 109*b14 + 84*b13 + 21*b12 + 109*b11 - 84*b10 + 19*b7 - 19*b6 + 62*b5 + 171*b4 - 41*b2 + 74*b1 - 155 $$\nu^{7}$$ $$=$$ $$( 2184 \beta_{15} - 111 \beta_{14} - 503 \beta_{13} - 346 \beta_{12} + 2295 \beta_{11} + 1681 \beta_{10} + 3096 \beta_{9} - 692 \beta_{8} + 648 \beta_{7} - 44 \beta_{6} - 1419 \beta_{5} - 3972 \beta_{4} - 1742 \beta_{3} + \cdots + 677 ) / 5$$ (2184*b15 - 111*b14 - 503*b13 - 346*b12 + 2295*b11 + 1681*b10 + 3096*b9 - 692*b8 + 648*b7 - 44*b6 - 1419*b5 - 3972*b4 - 1742*b3 - 1092*b2 - 2074*b1 + 677) / 5 $$\nu^{8}$$ $$=$$ $$- 1082 \beta_{14} - 820 \beta_{13} - 194 \beta_{12} - 1082 \beta_{11} + 820 \beta_{10} - 237 \beta_{7} + 237 \beta_{6} - 719 \beta_{5} - 1801 \beta_{4} + 460 \beta_{2} - 849 \beta _1 + 1509$$ -1082*b14 - 820*b13 - 194*b12 - 1082*b11 + 820*b10 - 237*b7 + 237*b6 - 719*b5 - 1801*b4 + 460*b2 - 849*b1 + 1509 $$\nu^{9}$$ $$=$$ $$( - 21824 \beta_{15} + 1359 \beta_{14} + 4493 \beta_{13} + 3576 \beta_{12} - 23183 \beta_{11} - 17331 \beta_{10} - 31024 \beta_{9} + 7152 \beta_{8} - 6143 \beta_{7} + 1009 \beta_{6} + 13919 \beta_{5} + \cdots - 7385 ) / 5$$ (-21824*b15 + 1359*b14 + 4493*b13 + 3576*b12 - 23183*b11 - 17331*b10 - 31024*b9 + 7152*b8 - 6143*b7 + 1009*b6 + 13919*b5 + 40288*b4 + 16254*b3 + 10912*b2 + 20398*b1 - 7385) / 5 $$\nu^{10}$$ $$=$$ $$10857 \beta_{14} + 8194 \beta_{13} + 1830 \beta_{12} + 10857 \beta_{11} - 8194 \beta_{10} + 2603 \beta_{7} - 2603 \beta_{6} + 7666 \beta_{5} + 18523 \beta_{4} - 4864 \beta_{2} + 9022 \beta _1 - 15185$$ 10857*b14 + 8194*b13 + 1830*b12 + 10857*b11 - 8194*b10 + 2603*b7 - 2603*b6 + 7666*b5 + 18523*b4 - 4864*b2 + 9022*b1 - 15185 $$\nu^{11}$$ $$=$$ $$( 220506 \beta_{15} - 14974 \beta_{14} - 43387 \beta_{13} - 36754 \beta_{12} + 235480 \beta_{11} + 177119 \beta_{10} + 313664 \beta_{9} - 73508 \beta_{8} + 60927 \beta_{7} - 12581 \beta_{6} + \cdots + 77143 ) / 5$$ (220506*b15 - 14974*b14 - 43387*b13 - 36754*b12 + 235480*b11 + 177119*b10 + 313664*b9 - 73508*b8 + 60927*b7 - 12581*b6 - 139891*b5 - 409253*b4 - 159378*b3 - 110253*b2 - 204916*b1 + 77143) / 5 $$\nu^{12}$$ $$=$$ $$- 109708 \beta_{14} - 82728 \beta_{13} - 17847 \beta_{12} - 109708 \beta_{11} + 82728 \beta_{10} - 27286 \beta_{7} + 27286 \beta_{6} - 79388 \beta_{5} - 189096 \beta_{4} + 50276 \beta_{2} + \cdots + 154077$$ -109708*b14 - 82728*b13 - 17847*b12 - 109708*b11 + 82728*b10 - 27286*b7 + 27286*b6 - 79388*b5 - 189096*b4 + 50276*b2 - 93299*b1 + 154077 $$\nu^{13}$$ $$=$$ $$( - 2236846 \beta_{15} + 157931 \beta_{14} + 432317 \beta_{13} + 375854 \beta_{12} - 2394777 \beta_{11} - 1804529 \beta_{10} - 3182446 \beta_{9} + 751708 \beta_{8} - 614142 \beta_{7} + \cdots - 792740 ) / 5$$ (-2236846*b15 + 157931*b14 + 432317*b13 + 375854*b12 - 2394777*b11 - 1804529*b10 - 3182446*b9 + 751708*b8 - 614142*b7 + 137566*b6 + 1416976*b5 + 4160247*b4 + 1596966*b3 + 1118423*b2 + 2074837*b1 - 792740) / 5 $$\nu^{14}$$ $$=$$ $$1112529 \beta_{14} + 838847 \beta_{13} + 177854 \beta_{12} + 1112529 \beta_{11} - 838847 \beta_{10} + 280875 \beta_{7} - 280875 \beta_{6} + 813203 \beta_{5} + 1925732 \beta_{4} + \cdots - 1566411$$ 1112529*b14 + 838847*b13 + 177854*b12 + 1112529*b11 - 838847*b10 + 280875*b7 - 280875*b6 + 813203*b5 + 1925732*b4 - 514819*b2 + 955029*b1 - 1566411 $$\nu^{15}$$ $$=$$ $$( 22725724 \beta_{15} - 1632836 \beta_{14} - 4361108 \beta_{13} - 3832591 \beta_{12} + 24358560 \beta_{11} + 18364616 \beta_{10} + 32334926 \beta_{9} - 7665182 \beta_{8} + \cdots + 8095157 ) / 5$$ (22725724*b15 - 1632836*b14 - 4361108*b13 - 3832591*b12 + 24358560*b11 + 18364616*b10 + 32334926*b9 - 7665182*b8 + 6226073*b7 - 1439109*b6 - 14390294*b5 - 42303192*b4 - 16144612*b3 - 11362862*b2 - 21068004*b1 + 8095157) / 5

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 + \beta_{4} + \beta_{5} - \beta_{9}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
126.1
 0.991969i 3.18910i − 1.20005i − 1.80544i 1.63097i − 1.51514i − 2.04679i 0.0288455i − 1.63097i 1.51514i 2.04679i − 0.0288455i − 0.991969i − 3.18910i 1.20005i 1.80544i
−0.520202 1.60102i −0.574677 + 0.417528i −0.674615 + 0.490137i 0 0.967418 + 0.702870i 4.59110 −1.58816 1.15387i −0.771126 + 2.37328i 0
126.2 0.154814 + 0.476469i −2.50250 + 1.81817i 1.41498 1.02804i 0 −1.25372 0.910884i 0.0237879 1.51951 + 1.10399i 2.02970 6.24676i 0
126.3 0.718805 + 2.21225i 1.86261 1.35327i −2.75935 + 2.00479i 0 4.33262 + 3.14783i 3.59425 −2.65482 1.92884i 0.710939 2.18805i 0
126.4 0.764617 + 2.35325i −1.71249 + 1.24419i −3.33511 + 2.42310i 0 −4.23730 3.07858i −0.973070 −4.24866 3.08683i 0.457541 1.40817i 0
251.1 −2.15604 + 1.56645i 0.234569 0.721930i 1.57669 4.85257i 0 0.625130 + 1.92395i −2.04213 2.55484 + 7.86300i 1.96089 + 1.42467i 0
251.2 −0.855434 + 0.621509i −0.212419 + 0.653760i −0.272540 + 0.838792i 0 −0.224607 0.691269i 1.01199 −0.941671 2.89816i 2.04477 + 1.48561i 0
251.3 0.264347 0.192059i −0.530081 + 1.63142i −0.585041 + 1.80057i 0 0.173205 + 0.533069i 3.42409 0.393106 + 1.20986i 0.0465016 + 0.0337854i 0
251.4 1.62909 1.18361i 0.934982 2.87758i 0.634989 1.95429i 0 −1.88274 5.79449i 0.369971 −0.0341417 0.105077i −4.97921 3.61761i 0
376.1 −2.15604 1.56645i 0.234569 + 0.721930i 1.57669 + 4.85257i 0 0.625130 1.92395i −2.04213 2.55484 7.86300i 1.96089 1.42467i 0
376.2 −0.855434 0.621509i −0.212419 0.653760i −0.272540 0.838792i 0 −0.224607 + 0.691269i 1.01199 −0.941671 + 2.89816i 2.04477 1.48561i 0
376.3 0.264347 + 0.192059i −0.530081 1.63142i −0.585041 1.80057i 0 0.173205 0.533069i 3.42409 0.393106 1.20986i 0.0465016 0.0337854i 0
376.4 1.62909 + 1.18361i 0.934982 + 2.87758i 0.634989 + 1.95429i 0 −1.88274 + 5.79449i 0.369971 −0.0341417 + 0.105077i −4.97921 + 3.61761i 0
501.1 −0.520202 + 1.60102i −0.574677 0.417528i −0.674615 0.490137i 0 0.967418 0.702870i 4.59110 −1.58816 + 1.15387i −0.771126 2.37328i 0
501.2 0.154814 0.476469i −2.50250 1.81817i 1.41498 + 1.02804i 0 −1.25372 + 0.910884i 0.0237879 1.51951 1.10399i 2.02970 + 6.24676i 0
501.3 0.718805 2.21225i 1.86261 + 1.35327i −2.75935 2.00479i 0 4.33262 3.14783i 3.59425 −2.65482 + 1.92884i 0.710939 + 2.18805i 0
501.4 0.764617 2.35325i −1.71249 1.24419i −3.33511 2.42310i 0 −4.23730 + 3.07858i −0.973070 −4.24866 + 3.08683i 0.457541 + 1.40817i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 501.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.d.n 16
5.b even 2 1 625.2.d.p 16
5.c odd 4 2 625.2.e.k 32
25.d even 5 1 625.2.a.g yes 8
25.d even 5 2 625.2.d.m 16
25.d even 5 1 inner 625.2.d.n 16
25.e even 10 1 625.2.a.e 8
25.e even 10 1 625.2.d.p 16
25.e even 10 2 625.2.d.q 16
25.f odd 20 2 625.2.b.d 16
25.f odd 20 4 625.2.e.j 32
25.f odd 20 2 625.2.e.k 32
75.h odd 10 1 5625.2.a.be 8
75.j odd 10 1 5625.2.a.s 8
100.h odd 10 1 10000.2.a.bn 8
100.j odd 10 1 10000.2.a.be 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.e 8 25.e even 10 1
625.2.a.g yes 8 25.d even 5 1
625.2.b.d 16 25.f odd 20 2
625.2.d.m 16 25.d even 5 2
625.2.d.n 16 1.a even 1 1 trivial
625.2.d.n 16 25.d even 5 1 inner
625.2.d.p 16 5.b even 2 1
625.2.d.p 16 25.e even 10 1
625.2.d.q 16 25.e even 10 2
625.2.e.j 32 25.f odd 20 4
625.2.e.k 32 5.c odd 4 2
625.2.e.k 32 25.f odd 20 2
5625.2.a.s 8 75.j odd 10 1
5625.2.a.be 8 75.h odd 10 1
10000.2.a.be 8 100.j odd 10 1
10000.2.a.bn 8 100.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 $$S_{2}^{\mathrm{new}}(625, [\chi])$$:

 $$T_{2}^{16} + 8 T_{2}^{14} + 10 T_{2}^{13} + 48 T_{2}^{12} - 20 T_{2}^{11} + 336 T_{2}^{10} - 185 T_{2}^{9} + 1045 T_{2}^{8} + 60 T_{2}^{7} + 2406 T_{2}^{6} + 1665 T_{2}^{5} + 1278 T_{2}^{4} - 1080 T_{2}^{3} + 1053 T_{2}^{2} - 405 T_{2} + 81$$ T2^16 + 8*T2^14 + 10*T2^13 + 48*T2^12 - 20*T2^11 + 336*T2^10 - 185*T2^9 + 1045*T2^8 + 60*T2^7 + 2406*T2^6 + 1665*T2^5 + 1278*T2^4 - 1080*T2^3 + 1053*T2^2 - 405*T2 + 81 $$T_{3}^{16} + 5 T_{3}^{15} + 17 T_{3}^{14} + 50 T_{3}^{13} + 168 T_{3}^{12} + 295 T_{3}^{11} + 529 T_{3}^{10} + 1255 T_{3}^{9} + 4220 T_{3}^{8} + 9055 T_{3}^{7} + 15869 T_{3}^{6} + 16150 T_{3}^{5} + 14803 T_{3}^{4} + 10330 T_{3}^{3} + \cdots + 841$$ T3^16 + 5*T3^15 + 17*T3^14 + 50*T3^13 + 168*T3^12 + 295*T3^11 + 529*T3^10 + 1255*T3^9 + 4220*T3^8 + 9055*T3^7 + 15869*T3^6 + 16150*T3^5 + 14803*T3^4 + 10330*T3^3 + 6327*T3^2 + 2610*T3 + 841

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 8 T^{14} + 10 T^{13} + 48 T^{12} + \cdots + 81$$
$3$ $$T^{16} + 5 T^{15} + 17 T^{14} + 50 T^{13} + \cdots + 841$$
$5$ $$T^{16}$$
$7$ $$(T^{8} - 10 T^{7} + 24 T^{6} + 35 T^{5} + \cdots + 1)^{2}$$
$11$ $$T^{16} - 2 T^{15} + 30 T^{14} + \cdots + 5861241$$
$13$ $$T^{16} + 5 T^{15} + 17 T^{14} + \cdots + 130321$$
$17$ $$T^{16} - 10 T^{15} + 88 T^{14} + \cdots + 2595321$$
$19$ $$T^{16} + 20 T^{14} + \cdots + 110775625$$
$23$ $$T^{16} + 15 T^{15} + \cdots + 2124195921$$
$29$ $$T^{16} + 10 T^{15} + \cdots + 3717950625$$
$31$ $$T^{16} - 17 T^{15} + \cdots + 576048001$$
$37$ $$T^{16} - 15 T^{15} + 83 T^{14} + \cdots + 6561$$
$41$ $$T^{16} - 12 T^{15} + \cdots + 237782041641$$
$43$ $$(T^{8} - 99 T^{6} + 180 T^{5} + 1946 T^{4} + \cdots - 1949)^{2}$$
$47$ $$T^{16} + 25 T^{15} + \cdots + 3244555521$$
$53$ $$T^{16} + 17 T^{14} + 185 T^{13} + \cdots + 3606201$$
$59$ $$T^{16} - 20 T^{15} + 225 T^{14} + \cdots + 50625$$
$61$ $$T^{16} + 23 T^{15} + \cdots + 10718253841$$
$67$ $$T^{16} + 208 T^{14} + \cdots + 3049560197401$$
$71$ $$T^{16} - 22 T^{15} + \cdots + 280529001$$
$73$ $$T^{16} + 40 T^{15} + \cdots + 56212142281$$
$79$ $$T^{16} - 75 T^{15} + \cdots + 62262725625$$
$83$ $$T^{16} + \cdots + 150377514648201$$
$89$ $$T^{16} - 5 T^{15} + \cdots + 3419818025625$$
$97$ $$T^{16} + 40 T^{15} + \cdots + 945602601241$$