# Properties

 Label 462.2.i.f Level $462$ Weight $2$ Character orbit 462.i Analytic conductor $3.689$ Analytic rank $0$ Dimension $6$ 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.1156923.1 Defining polynomial: $$x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4$$ x^6 - 3*x^5 + 12*x^4 - 19*x^3 + 27*x^2 - 18*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 3$$ v^2 - v + 3 $$\beta_{2}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 22\nu^{3} - 28\nu^{2} + 43\nu - 18 ) / 2$$ (2*v^5 - 5*v^4 + 22*v^3 - 28*v^2 + 43*v - 18) / 2 $$\beta_{3}$$ $$=$$ $$-3\nu^{5} + 7\nu^{4} - 31\nu^{3} + 37\nu^{2} - 56\nu + 22$$ -3*v^5 + 7*v^4 - 31*v^3 + 37*v^2 - 56*v + 22 $$\beta_{4}$$ $$=$$ $$( -6\nu^{5} + 15\nu^{4} - 64\nu^{3} + 82\nu^{2} - 121\nu + 50 ) / 2$$ (-6*v^5 + 15*v^4 - 64*v^3 + 82*v^2 - 121*v + 50) / 2 $$\beta_{5}$$ $$=$$ $$-3\nu^{5} + 8\nu^{4} - 33\nu^{3} + 44\nu^{2} - 62\nu + 24$$ -3*v^5 + 8*v^4 - 33*v^3 + 44*v^2 - 62*v + 24
 $$\nu$$ $$=$$ $$( \beta_{5} - 2\beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2$$ (b5 - 2*b4 + b3 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - 2\beta_{4} + \beta_{3} + 3\beta _1 - 5 ) / 2$$ (b5 - 2*b4 + b3 + 3*b1 - 5) / 2 $$\nu^{3}$$ $$=$$ $$( -3\beta_{5} + 8\beta_{4} - 3\beta_{3} + 6\beta_{2} - \beta _1 - 5 ) / 2$$ (-3*b5 + 8*b4 - 3*b3 + 6*b2 - b1 - 5) / 2 $$\nu^{4}$$ $$=$$ $$( -5\beta_{5} + 18\beta_{4} - 9\beta_{3} + 12\beta_{2} - 17\beta _1 + 27 ) / 2$$ (-5*b5 + 18*b4 - 9*b3 + 12*b2 - 17*b1 + 27) / 2 $$\nu^{5}$$ $$=$$ $$( 13\beta_{5} - 28\beta_{4} + 3\beta_{3} - 34\beta_{2} - 11\beta _1 + 49 ) / 2$$ (13*b5 - 28*b4 + 3*b3 - 34*b2 - 11*b1 + 49) / 2

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.0585812i 0.5 + 1.51496i 0.5 − 2.43956i 0.5 − 0.0585812i 0.5 − 1.51496i 0.5 + 2.43956i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.37328 2.37860i 1.00000 −1.37328 + 2.26144i 1.00000 −0.500000 0.866025i −1.37328 + 2.37860i
67.2 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.227452 0.393958i 1.00000 −0.227452 2.63596i 1.00000 −0.500000 0.866025i −0.227452 + 0.393958i
67.3 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.60074 + 2.77256i 1.00000 1.60074 + 2.10657i 1.00000 −0.500000 0.866025i 1.60074 2.77256i
331.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.37328 + 2.37860i 1.00000 −1.37328 2.26144i 1.00000 −0.500000 + 0.866025i −1.37328 2.37860i
331.2 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.227452 + 0.393958i 1.00000 −0.227452 + 2.63596i 1.00000 −0.500000 + 0.866025i −0.227452 0.393958i
331.3 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.60074 2.77256i 1.00000 1.60074 2.10657i 1.00000 −0.500000 + 0.866025i 1.60074 + 2.77256i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 331.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.i.f 6
3.b odd 2 1 1386.2.k.w 6
7.c even 3 1 inner 462.2.i.f 6
7.c even 3 1 3234.2.a.bi 3
7.d odd 6 1 3234.2.a.bg 3
21.g even 6 1 9702.2.a.du 3
21.h odd 6 1 1386.2.k.w 6
21.h odd 6 1 9702.2.a.dt 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.f 6 1.a even 1 1 trivial
462.2.i.f 6 7.c even 3 1 inner
1386.2.k.w 6 3.b odd 2 1
1386.2.k.w 6 21.h odd 6 1
3234.2.a.bg 3 7.d odd 6 1
3234.2.a.bi 3 7.c even 3 1
9702.2.a.dt 3 21.h odd 6 1
9702.2.a.du 3 21.g even 6 1

## Hecke kernels

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

 $$T_{5}^{6} + 9T_{5}^{4} + 8T_{5}^{3} + 81T_{5}^{2} + 36T_{5} + 16$$ T5^6 + 9*T5^4 + 8*T5^3 + 81*T5^2 + 36*T5 + 16 $$T_{13}^{3} - 36T_{13} + 32$$ T13^3 - 36*T13 + 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{3}$$
$3$ $$(T^{2} + T + 1)^{3}$$
$5$ $$T^{6} + 9 T^{4} + 8 T^{3} + 81 T^{2} + \cdots + 16$$
$7$ $$T^{6} + 12 T^{4} - 4 T^{3} + 84 T^{2} + \cdots + 343$$
$11$ $$(T^{2} + T + 1)^{3}$$
$13$ $$(T^{3} - 36 T + 32)^{2}$$
$17$ $$(T^{2} + T + 1)^{3}$$
$19$ $$T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 1296$$
$23$ $$T^{6} + 9 T^{5} + 78 T^{4} + 69 T^{3} + \cdots + 441$$
$29$ $$(T^{3} - 9 T^{2} + 92)^{2}$$
$31$ $$T^{6} + 6 T^{5} + 132 T^{4} + \cdots + 262144$$
$37$ $$T^{6} + 9 T^{5} + 93 T^{4} + \cdots + 26896$$
$41$ $$(T^{3} - 12 T^{2} + 21 T + 82)^{2}$$
$43$ $$(T^{3} - 9 T^{2} - 12 T + 164)^{2}$$
$47$ $$T^{6} - 15 T^{5} + 174 T^{4} + \cdots + 361$$
$53$ $$T^{6} + 36 T^{4} + 64 T^{3} + \cdots + 1024$$
$59$ $$T^{6} + 9 T^{5} + 177 T^{4} + \cdots + 589824$$
$61$ $$T^{6} - 6 T^{5} + 165 T^{4} + \cdots + 51076$$
$67$ $$T^{6} + 6 T^{5} + 51 T^{4} + \cdots + 7056$$
$71$ $$(T^{3} - 3 T^{2} - 24 T + 64)^{2}$$
$73$ $$T^{6} + 96 T^{4} - 384 T^{3} + \cdots + 36864$$
$79$ $$T^{6} + 12 T^{5} + 177 T^{4} + \cdots + 135424$$
$83$ $$(T^{3} + 6 T^{2} - 111 T + 188)^{2}$$
$89$ $$T^{6} - 36 T^{5} + 900 T^{4} + \cdots + 1763584$$
$97$ $$(T^{3} - 9 T^{2} - 81 T - 7)^{2}$$