# Properties

 Label 462.2.k.d Level $462$ Weight $2$ Character orbit 462.k Analytic conductor $3.689$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.k (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ 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: $$2^{2}\cdot 3^{2}$$ 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.

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

Basis of coefficient ring

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

## Character values

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

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$-1$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i
−0.866025 0.500000i −1.50000 + 0.866025i 0.500000 + 0.866025i −0.358719 + 0.621320i 1.73205 2.62132 + 0.358719i 1.00000i 1.50000 2.59808i 0.621320 0.358719i
89.2 −0.866025 0.500000i −1.50000 + 0.866025i 0.500000 + 0.866025i 2.09077 3.62132i 1.73205 −1.62132 2.09077i 1.00000i 1.50000 2.59808i −3.62132 + 2.09077i
89.3 0.866025 + 0.500000i −1.50000 + 0.866025i 0.500000 + 0.866025i −2.09077 + 3.62132i −1.73205 −1.62132 2.09077i 1.00000i 1.50000 2.59808i −3.62132 + 2.09077i
89.4 0.866025 + 0.500000i −1.50000 + 0.866025i 0.500000 + 0.866025i 0.358719 0.621320i −1.73205 2.62132 + 0.358719i 1.00000i 1.50000 2.59808i 0.621320 0.358719i
353.1 −0.866025 + 0.500000i −1.50000 0.866025i 0.500000 0.866025i −0.358719 0.621320i 1.73205 2.62132 0.358719i 1.00000i 1.50000 + 2.59808i 0.621320 + 0.358719i
353.2 −0.866025 + 0.500000i −1.50000 0.866025i 0.500000 0.866025i 2.09077 + 3.62132i 1.73205 −1.62132 + 2.09077i 1.00000i 1.50000 + 2.59808i −3.62132 2.09077i
353.3 0.866025 0.500000i −1.50000 0.866025i 0.500000 0.866025i −2.09077 3.62132i −1.73205 −1.62132 + 2.09077i 1.00000i 1.50000 + 2.59808i −3.62132 2.09077i
353.4 0.866025 0.500000i −1.50000 0.866025i 0.500000 0.866025i 0.358719 + 0.621320i −1.73205 2.62132 0.358719i 1.00000i 1.50000 + 2.59808i 0.621320 + 0.358719i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 353.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.k.d 8
3.b odd 2 1 inner 462.2.k.d 8
7.d odd 6 1 inner 462.2.k.d 8
21.g even 6 1 inner 462.2.k.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.k.d 8 1.a even 1 1 trivial
462.2.k.d 8 3.b odd 2 1 inner
462.2.k.d 8 7.d odd 6 1 inner
462.2.k.d 8 21.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$:

 $$T_{5}^{8} + 18T_{5}^{6} + 315T_{5}^{4} + 162T_{5}^{2} + 81$$ T5^8 + 18*T5^6 + 315*T5^4 + 162*T5^2 + 81 $$T_{17}^{8} + 54T_{17}^{6} + 2475T_{17}^{4} + 23814T_{17}^{2} + 194481$$ T17^8 + 54*T17^6 + 2475*T17^4 + 23814*T17^2 + 194481

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{2}$$
$3$ $$(T^{2} + 3 T + 3)^{4}$$
$5$ $$T^{8} + 18 T^{6} + 315 T^{4} + \cdots + 81$$
$7$ $$(T^{4} - 2 T^{3} - 3 T^{2} - 14 T + 49)^{2}$$
$11$ $$(T^{4} - T^{2} + 1)^{2}$$
$13$ $$(T^{2} + 12)^{4}$$
$17$ $$T^{8} + 54 T^{6} + 2475 T^{4} + \cdots + 194481$$
$19$ $$(T^{4} + 12 T^{3} + 36 T^{2} - 144 T + 144)^{2}$$
$23$ $$T^{8} - 54 T^{6} + 2835 T^{4} + \cdots + 6561$$
$29$ $$(T^{2} + 72)^{4}$$
$31$ $$(T^{4} + 12 T^{3} + 36 T^{2} - 144 T + 144)^{2}$$
$37$ $$(T^{2} + 4 T + 16)^{4}$$
$41$ $$(T^{2} - 27)^{4}$$
$43$ $$(T^{2} - 8 T - 56)^{4}$$
$47$ $$T^{8} + 18 T^{6} + 315 T^{4} + \cdots + 81$$
$53$ $$(T^{4} - 72 T^{2} + 5184)^{2}$$
$59$ $$T^{8} + 216 T^{6} + \cdots + 49787136$$
$61$ $$(T^{4} + 6 T^{3} + 9 T^{2} - 18 T + 9)^{2}$$
$67$ $$(T^{4} + 10 T^{3} + 147 T^{2} - 470 T + 2209)^{2}$$
$71$ $$(T^{2} + 36)^{4}$$
$73$ $$(T^{4} + 12 T^{3} + 36 T^{2} - 144 T + 144)^{2}$$
$79$ $$(T^{4} - 2 T^{3} + 21 T^{2} + 34 T + 289)^{2}$$
$83$ $$(T^{4} - 54 T^{2} + 441)^{2}$$
$89$ $$T^{8} + 144 T^{6} + 20160 T^{4} + \cdots + 331776$$
$97$ $$(T^{4} + 102 T^{2} + 9)^{2}$$