# Properties

 Label 775.2.j.a Level $775$ Weight $2$ Character orbit 775.j Analytic conductor $6.188$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$775 = 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 775.j (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.18840615665$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$$.

 $$n$$ $$251$$ $$652$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
156.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i
0.618034 + 1.90211i −2.61803 + 1.90211i −1.61803 + 1.17557i −0.690983 + 2.12663i −5.23607 3.80423i −0.236068 0 2.30902 7.10642i −4.47214
311.1 −1.61803 + 1.17557i −0.381966 + 1.17557i 0.618034 1.90211i −1.80902 1.31433i −0.763932 2.35114i 4.23607 0 1.19098 + 0.865300i 4.47214
466.1 −1.61803 1.17557i −0.381966 1.17557i 0.618034 + 1.90211i −1.80902 + 1.31433i −0.763932 + 2.35114i 4.23607 0 1.19098 0.865300i 4.47214
621.1 0.618034 1.90211i −2.61803 1.90211i −1.61803 1.17557i −0.690983 2.12663i −5.23607 + 3.80423i −0.236068 0 2.30902 + 7.10642i −4.47214
 $$n$$: e.g. 2-40 or 990-1000 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 775.2.j.a 4
25.d even 5 1 inner 775.2.j.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.j.a 4 1.a even 1 1 trivial
775.2.j.a 4 25.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 2T_{2}^{3} + 4T_{2}^{2} + 8T_{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(775, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$3$ $$T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16$$
$5$ $$T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25$$
$7$ $$(T^{2} - 4 T - 1)^{2}$$
$11$ $$T^{4} + 7 T^{3} + 34 T^{2} + 88 T + 121$$
$13$ $$T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121$$
$17$ $$T^{4} + 7 T^{3} + 19 T^{2} + 3 T + 1$$
$19$ $$T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 400$$
$23$ $$T^{4} + 11 T^{3} + 51 T^{2} + \cdots + 961$$
$29$ $$T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25$$
$31$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$37$ $$T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121$$
$41$ $$T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121$$
$43$ $$(T^{2} - 2 T - 19)^{2}$$
$47$ $$T^{4} + 22 T^{3} + 244 T^{2} + \cdots + 5776$$
$53$ $$T^{4} - 24 T^{3} + 306 T^{2} + \cdots + 9801$$
$59$ $$T^{4} - 15 T^{3} + 115 T^{2} + \cdots + 9025$$
$61$ $$T^{4} + 7 T^{3} + 199 T^{2} + \cdots + 361$$
$67$ $$T^{4} + 22 T^{3} + 304 T^{2} + \cdots + 10201$$
$71$ $$T^{4} + 2 T^{3} + 24 T^{2} + 133 T + 361$$
$73$ $$T^{4} + 16 T^{3} + 106 T^{2} + \cdots + 3721$$
$79$ $$T^{4} + 15 T^{3} + 100 T^{2} + \cdots + 625$$
$83$ $$T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121$$
$89$ $$T^{4} - 25 T^{3} + 235 T^{2} + \cdots + 3025$$
$97$ $$T^{4} + 12 T^{3} + 94 T^{2} + \cdots + 7921$$