# Properties

 Label 570.2.q.a Level $570$ Weight $2$ Character orbit 570.q Analytic conductor $4.551$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 570.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{5} + \zeta_{24}$$ v^5 + v $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{5}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}^{3}$$ v^7 + v^3 $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{5} + 2\zeta_{24}$$ -v^5 + 2*v $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{7} + 2\zeta_{24}^{3}$$ -v^7 + 2*v^3
 $$\zeta_{24}$$ $$=$$ $$( \beta_{6} + \beta_{3} ) / 3$$ (b6 + b3) / 3 $$\zeta_{24}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{5} ) / 3$$ (b7 + b5) / 3 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{6} + 2\beta_{3} ) / 3$$ (-b6 + 2*b3) / 3 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{4}$$ b4 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{7} + 2\beta_{5} ) / 3$$ (-b7 + 2*b5) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/570\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$211$$ $$457$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i
−0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −2.03906 0.917738i 0.500000 + 0.866025i 3.00000i 1.00000i 0.500000 + 0.866025i 1.30701 + 1.81431i
49.2 −0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 1.30701 1.81431i 0.500000 + 0.866025i 3.00000i 1.00000i 0.500000 + 0.866025i −2.03906 + 0.917738i
49.3 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0.917738 2.03906i 0.500000 + 0.866025i 3.00000i 1.00000i 0.500000 + 0.866025i 1.81431 1.30701i
49.4 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 1.81431 + 1.30701i 0.500000 + 0.866025i 3.00000i 1.00000i 0.500000 + 0.866025i 0.917738 + 2.03906i
349.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −2.03906 + 0.917738i 0.500000 0.866025i 3.00000i 1.00000i 0.500000 0.866025i 1.30701 1.81431i
349.2 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 1.30701 + 1.81431i 0.500000 0.866025i 3.00000i 1.00000i 0.500000 0.866025i −2.03906 0.917738i
349.3 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0.917738 + 2.03906i 0.500000 0.866025i 3.00000i 1.00000i 0.500000 0.866025i 1.81431 + 1.30701i
349.4 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 1.81431 1.30701i 0.500000 0.866025i 3.00000i 1.00000i 0.500000 0.866025i 0.917738 2.03906i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.q.a 8
3.b odd 2 1 1710.2.t.a 8
5.b even 2 1 inner 570.2.q.a 8
15.d odd 2 1 1710.2.t.a 8
19.c even 3 1 inner 570.2.q.a 8
57.h odd 6 1 1710.2.t.a 8
95.i even 6 1 inner 570.2.q.a 8
285.n odd 6 1 1710.2.t.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.q.a 8 1.a even 1 1 trivial
570.2.q.a 8 5.b even 2 1 inner
570.2.q.a 8 19.c even 3 1 inner
570.2.q.a 8 95.i even 6 1 inner
1710.2.t.a 8 3.b odd 2 1
1710.2.t.a 8 15.d odd 2 1
1710.2.t.a 8 57.h odd 6 1
1710.2.t.a 8 285.n odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{2}$$
$3$ $$(T^{4} - T^{2} + 1)^{2}$$
$5$ $$T^{8} - 4 T^{7} + 8 T^{6} + 8 T^{5} + \cdots + 625$$
$7$ $$(T^{2} + 9)^{4}$$
$11$ $$(T - 2)^{8}$$
$13$ $$T^{8} - 14 T^{6} + 171 T^{4} + \cdots + 625$$
$17$ $$(T^{4} - 24 T^{2} + 576)^{2}$$
$19$ $$(T^{2} - 2 T + 19)^{4}$$
$23$ $$(T^{4} - 6 T^{2} + 36)^{2}$$
$29$ $$(T^{4} + 4 T^{3} + 66 T^{2} - 200 T + 2500)^{2}$$
$31$ $$(T^{2} + 6 T - 15)^{4}$$
$37$ $$(T^{4} + 110 T^{2} + 1849)^{2}$$
$41$ $$(T^{4} + 6 T^{2} + 36)^{2}$$
$43$ $$T^{8} - 30 T^{6} + 891 T^{4} + \cdots + 81$$
$47$ $$T^{8} - 84 T^{6} + 6156 T^{4} + \cdots + 810000$$
$53$ $$T^{8} - 176 T^{6} + 29376 T^{4} + \cdots + 2560000$$
$59$ $$(T^{4} - 4 T^{3} + 108 T^{2} + 368 T + 8464)^{2}$$
$61$ $$(T^{4} - 6 T^{3} + 81 T^{2} + 270 T + 2025)^{2}$$
$67$ $$T^{8} - 14 T^{6} + 171 T^{4} + \cdots + 625$$
$71$ $$(T^{4} - 12 T^{3} + 114 T^{2} - 360 T + 900)^{2}$$
$73$ $$T^{8} - 146 T^{6} + 20691 T^{4} + \cdots + 390625$$
$79$ $$(T^{4} - 14 T^{3} + 243 T^{2} + 658 T + 2209)^{2}$$
$83$ $$(T^{4} + 180 T^{2} + 324)^{2}$$
$89$ $$(T^{4} + 28 T^{3} + 594 T^{2} + \cdots + 36100)^{2}$$
$97$ $$T^{8} - 464 T^{6} + \cdots + 1600000000$$