Properties

 Label 462.2.j.e Level $462$ Weight $2$ Character orbit 462.j Analytic conductor $3.689$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.j (of order $$5$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.484000000.9 Defining polynomial: $$x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121$$ x^8 + x^6 + 16*x^4 + 66*x^2 + 121 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 7\nu^{6} - 37\nu^{4} + 629\nu^{2} - 363 ) / 1991$$ (7*v^6 - 37*v^4 + 629*v^2 - 363) / 1991 $$\beta_{3}$$ $$=$$ $$( -28\nu^{6} + 148\nu^{4} - 525\nu^{2} - 539 ) / 1991$$ (-28*v^6 + 148*v^4 - 525*v^2 - 539) / 1991 $$\beta_{4}$$ $$=$$ $$( -28\nu^{7} + 148\nu^{5} - 525\nu^{3} - 539\nu ) / 1991$$ (-28*v^7 + 148*v^5 - 525*v^3 - 539*v) / 1991 $$\beta_{5}$$ $$=$$ $$( 40\nu^{6} + 73\nu^{4} + 750\nu^{2} + 2761 ) / 1991$$ (40*v^6 + 73*v^4 + 750*v^2 + 2761) / 1991 $$\beta_{6}$$ $$=$$ $$( -61\nu^{7} + 38\nu^{5} - 646\nu^{3} - 1672\nu ) / 1991$$ (-61*v^7 + 38*v^5 - 646*v^3 - 1672*v) / 1991 $$\beta_{7}$$ $$=$$ $$( 68\nu^{7} - 75\nu^{5} + 1275\nu^{3} + 3300\nu ) / 1991$$ (68*v^7 - 75*v^5 + 1275*v^3 + 3300*v) / 1991
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + 1$$ b3 + 4*b2 + 1 $$\nu^{3}$$ $$=$$ $$4\beta_{7} + 4\beta_{6} + \beta_{4} - 3\beta_1$$ 4*b7 + 4*b6 + b4 - 3*b1 $$\nu^{4}$$ $$=$$ $$7\beta_{5} + 10\beta_{3} - 7$$ 7*b5 + 10*b3 - 7 $$\nu^{5}$$ $$=$$ $$7\beta_{7} + 17\beta_{4} - 7\beta_1$$ 7*b7 + 17*b4 - 7*b1 $$\nu^{6}$$ $$=$$ $$37\beta_{5} - 37\beta_{3} - 75\beta_{2} - 75$$ 37*b5 - 37*b3 - 75*b2 - 75 $$\nu^{7}$$ $$=$$ $$-38\beta_{7} - 75\beta_{6}$$ -38*b7 - 75*b6

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/462\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{3}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 −0.476925 + 1.46782i 0.476925 − 1.46782i 1.73855 − 1.26313i −1.73855 + 1.26313i 1.73855 + 1.26313i −1.73855 − 1.26313i −0.476925 − 1.46782i 0.476925 + 1.46782i
−0.809017 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i −1.52029 + 1.10455i −0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i 1.87918
169.2 −0.809017 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i 2.52029 1.83110i −0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i −3.11525
295.1 0.309017 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0.0895849 + 0.275714i 0.309017 + 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i 0.289903
295.2 0.309017 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0.910415 + 2.80197i 0.309017 + 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i 2.94617
379.1 0.309017 + 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0.0895849 0.275714i 0.309017 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i 0.289903
379.2 0.309017 + 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0.910415 2.80197i 0.309017 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i 2.94617
421.1 −0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i −1.52029 1.10455i −0.809017 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i 1.87918
421.2 −0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i 2.52029 + 1.83110i −0.809017 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i −3.11525
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 421.2 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 462.2.j.e 8
11.c even 5 1 inner 462.2.j.e 8
11.c even 5 1 5082.2.a.cf 4
11.d odd 10 1 5082.2.a.ca 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.j.e 8 1.a even 1 1 trivial
462.2.j.e 8 11.c even 5 1 inner
5082.2.a.ca 4 11.d odd 10 1
5082.2.a.cf 4 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 4T_{5}^{7} + 11T_{5}^{6} - 4T_{5}^{5} - 4T_{5}^{4} + 40T_{5}^{3} + 290T_{5}^{2} - 50T_{5} + 25$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$3$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$5$ $$T^{8} - 4 T^{7} + 11 T^{6} - 4 T^{5} + \cdots + 25$$
$7$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$11$ $$T^{8} - 29 T^{6} + 451 T^{4} + \cdots + 14641$$
$13$ $$T^{8} - 4 T^{7} + 18 T^{6} - 20 T^{5} + \cdots + 16$$
$17$ $$T^{8} + 8 T^{7} + 104 T^{6} + \cdots + 24025$$
$19$ $$T^{8} + 12 T^{7} + 69 T^{6} + 208 T^{5} + \cdots + 25$$
$23$ $$(T^{4} + 2 T^{3} - 9 T^{2} + 5)^{2}$$
$29$ $$T^{8} + 4 T^{7} + 46 T^{6} + 304 T^{5} + \cdots + 400$$
$31$ $$T^{8} + 14 T^{7} + 117 T^{6} + \cdots + 19321$$
$37$ $$T^{8} - 10 T^{7} + 54 T^{6} + \cdots + 2401$$
$41$ $$T^{8} + 12 T^{7} + 44 T^{6} + \cdots + 3025$$
$43$ $$(T^{4} - 10 T^{3} - 72 T^{2} + 660 T + 116)^{2}$$
$47$ $$T^{8} - 28 T^{7} + 422 T^{6} + \cdots + 55696$$
$53$ $$T^{8} - 2 T^{7} - 2 T^{6} + \cdots + 55696$$
$59$ $$T^{8} - 10 T^{6} + 3400 T^{4} + \cdots + 250000$$
$61$ $$T^{8} + 34 T^{7} + 562 T^{6} + \cdots + 355216$$
$67$ $$(T^{4} + 12 T^{3} - 58 T^{2} - 564 T + 404)^{2}$$
$71$ $$T^{8} + 44 T^{6} + 240 T^{5} + \cdots + 8202496$$
$73$ $$T^{8} + 86 T^{6} + 600 T^{5} + \cdots + 839056$$
$79$ $$T^{8} - 22 T^{7} + 384 T^{6} + \cdots + 17808400$$
$83$ $$T^{8} + 30 T^{7} + \cdots + 133726096$$
$89$ $$(T^{4} + 2 T^{3} - 309 T^{2} - 530 T + 20725)^{2}$$
$97$ $$T^{8} + 10 T^{7} + 54 T^{6} + 160 T^{5} + \cdots + 16$$