# Properties

 Label 775.2.a.h Level $775$ Weight $2$ Character orbit 775.a Self dual yes Analytic conductor $6.188$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$775 = 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 775.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.18840615665$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.144209.1 Defining polynomial: $$x^{5} - 6x^{3} - x^{2} + 6x - 1$$ x^5 - 6*x^3 - x^2 + 6*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 6x^{3} - x^{2} + 6x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4\nu + 2$$ v^3 - v^2 - 4*v + 2 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 5\nu^{2} - 2\nu + 2$$ v^4 - 5*v^2 - 2*v + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta_1$$ b3 + b2 + 4*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5\beta_{2} + 2\beta _1 + 8$$ b4 + 5*b2 + 2*b1 + 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.871612 −1.93413 2.32352 0.177477 −1.43848
−2.58398 −2.24029 4.67698 0 5.78888 2.11190 −6.91727 2.01891 0
1.2 −2.23959 0.740841 3.01578 0 −1.65918 −3.67497 −2.27494 −2.45115 0
1.3 −1.14876 2.39873 −0.680351 0 −2.75556 −1.07521 3.07908 2.75390 0
1.4 0.264183 −2.96850 −1.93021 0 −0.784228 2.14598 −1.03829 5.81200 0
1.5 1.70816 −0.930775 0.917797 0 −1.58991 −1.50771 −1.84857 −2.13366 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$31$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.a.h 5
3.b odd 2 1 6975.2.a.by 5
5.b even 2 1 775.2.a.k yes 5
5.c odd 4 2 775.2.b.g 10
15.d odd 2 1 6975.2.a.bp 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.a.h 5 1.a even 1 1 trivial
775.2.a.k yes 5 5.b even 2 1
775.2.b.g 10 5.c odd 4 2
6975.2.a.bp 5 15.d odd 2 1
6975.2.a.by 5 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} + 4T_{2}^{4} - 13T_{2}^{2} - 8T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(775))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} + 4 T^{4} - 13 T^{2} - 8 T + 3$$
$3$ $$T^{5} + 3 T^{4} - 6 T^{3} - 19 T^{2} + \cdots + 11$$
$5$ $$T^{5}$$
$7$ $$T^{5} + 2 T^{4} - 11 T^{3} - 13 T^{2} + \cdots + 27$$
$11$ $$T^{5} + 2 T^{4} - 37 T^{3} - 41 T^{2} + \cdots - 233$$
$13$ $$T^{5} + 4 T^{4} - 25 T^{3} - 67 T^{2} + \cdots + 153$$
$17$ $$T^{5} + 19 T^{4} + 129 T^{3} + \cdots - 59$$
$19$ $$T^{5} - 8 T^{4} - 3 T^{3} + 94 T^{2} + \cdots + 59$$
$23$ $$T^{5} + 12 T^{4} + 3 T^{3} - 366 T^{2} + \cdots + 503$$
$29$ $$T^{5} - 6 T^{4} - 87 T^{3} + \cdots - 8711$$
$31$ $$(T + 1)^{5}$$
$37$ $$T^{5} + 16 T^{4} - 911 T^{2} + \cdots - 4369$$
$41$ $$T^{5} + 2 T^{4} - 127 T^{3} + \cdots - 507$$
$43$ $$T^{5} + T^{4} - 40 T^{3} - 43 T^{2} + \cdots + 17$$
$47$ $$T^{5} + 8 T^{4} - 91 T^{3} - 578 T^{2} + \cdots - 291$$
$53$ $$T^{5} + 3 T^{4} - 102 T^{3} + \cdots + 389$$
$59$ $$T^{5} + 4 T^{4} - 142 T^{3} + \cdots + 15079$$
$61$ $$T^{5} + 5 T^{4} - 29 T^{3} - 3 T^{2} + \cdots - 3$$
$67$ $$T^{5} + 13 T^{4} - 234 T^{3} + \cdots + 124659$$
$71$ $$T^{5} - 6 T^{4} - 279 T^{3} + \cdots + 57279$$
$73$ $$T^{5} + 19 T^{4} - 11 T^{3} + \cdots + 3291$$
$79$ $$T^{5} + 6 T^{4} - 84 T^{3} + \cdots + 1873$$
$83$ $$T^{5} + 18 T^{4} - 33 T^{3} + \cdots - 1983$$
$89$ $$T^{5} + 31 T^{4} + 361 T^{3} + \cdots + 729$$
$97$ $$T^{5} - T^{4} - 271 T^{3} + \cdots - 25961$$