# Properties

 Label 930.2.i.n Level $930$ Weight $2$ Character orbit 930.i Analytic conductor $7.426$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.3636603.4 Defining polynomial: $$x^{6} - x^{5} + 3x^{4} + 4x^{3} + 12x^{2} - 16x + 64$$ x^6 - x^5 + 3*x^4 + 4*x^3 + 12*x^2 - 16*x + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3x^{4} + 4x^{3} + 12x^{2} - 16x + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 3\nu^{4} + 7\nu^{3} - 2\nu^{2} + 8\nu + 32 ) / 64$$ (-v^5 + 3*v^4 + 7*v^3 - 2*v^2 + 8*v + 32) / 64 $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 5\nu^{4} + \nu^{3} - 22\nu^{2} - 24\nu + 32 ) / 64$$ (v^5 + 5*v^4 + v^3 - 22*v^2 - 24*v + 32) / 64 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} - 13\nu^{4} - 9\nu^{3} - 82\nu^{2} + 40\nu - 224 ) / 64$$ (-v^5 - 13*v^4 - 9*v^3 - 82*v^2 + 40*v - 224) / 64 $$\beta_{4}$$ $$=$$ $$( 7\nu^{5} + 3\nu^{4} + 7\nu^{3} + 38\nu^{2} + 120\nu - 32 ) / 64$$ (7*v^5 + 3*v^4 + 7*v^3 + 38*v^2 + 120*v - 32) / 64 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 3\nu^{3} - 4\nu^{2} + 4\nu + 8 ) / 8$$ (-v^5 + v^4 - 3*v^3 - 4*v^2 + 4*v + 8) / 8
 $$\nu$$ $$=$$ $$( \beta_{5} + 2\beta_{4} + \beta_{3} - 2\beta_{2} + 3\beta _1 + 3 ) / 6$$ (b5 + 2*b4 + b3 - 2*b2 + 3*b1 + 3) / 6 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} - 2\beta_{2} - \beta _1 - 2 ) / 2$$ (-b3 - 2*b2 - b1 - 2) / 2 $$\nu^{3}$$ $$=$$ $$( -7\beta_{5} - 2\beta_{4} + 2\beta_{3} - 4\beta_{2} + 36\beta _1 - 3 ) / 6$$ (-7*b5 - 2*b4 + 2*b3 - 4*b2 + 36*b1 - 3) / 6 $$\nu^{4}$$ $$=$$ $$( 3\beta_{5} + 2\beta_{4} - 3\beta_{3} + 10\beta_{2} + 3\beta _1 - 19 ) / 2$$ (3*b5 + 2*b4 - 3*b3 + 10*b2 + 3*b1 - 19) / 2 $$\nu^{5}$$ $$=$$ $$( -14\beta_{5} + 20\beta_{4} + \beta_{3} + 58\beta_{2} - 75\beta _1 + 36 ) / 6$$ (-14*b5 + 20*b4 + b3 + 58*b2 - 75*b1 + 36) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
211.1
 0.715814 + 1.86751i −1.55951 − 1.25217i 1.34370 − 1.48137i 0.715814 − 1.86751i −1.55951 + 1.25217i 1.34370 + 1.48137i
−1.00000 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i −2.25941 3.91341i −1.00000 −0.500000 + 0.866025i 0.500000 0.866025i
211.2 −1.00000 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i −0.695350 1.20438i −1.00000 −0.500000 + 0.866025i 0.500000 0.866025i
211.3 −1.00000 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i 0.954758 + 1.65369i −1.00000 −0.500000 + 0.866025i 0.500000 0.866025i
811.1 −1.00000 0.500000 0.866025i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i −2.25941 + 3.91341i −1.00000 −0.500000 0.866025i 0.500000 + 0.866025i
811.2 −1.00000 0.500000 0.866025i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i −0.695350 + 1.20438i −1.00000 −0.500000 0.866025i 0.500000 + 0.866025i
811.3 −1.00000 0.500000 0.866025i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i 0.954758 1.65369i −1.00000 −0.500000 0.866025i 0.500000 + 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 811.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.i.n 6
31.c even 3 1 inner 930.2.i.n 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.i.n 6 1.a even 1 1 trivial
930.2.i.n 6 31.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$:

 $$T_{7}^{6} + 4T_{7}^{5} + 21T_{7}^{4} + 4T_{7}^{3} + 73T_{7}^{2} + 60T_{7} + 144$$ T7^6 + 4*T7^5 + 21*T7^4 + 4*T7^3 + 73*T7^2 + 60*T7 + 144 $$T_{11}^{6} - 5T_{11}^{5} + 54T_{11}^{4} - 113T_{11}^{3} + 1486T_{11}^{2} - 3741T_{11} + 16641$$ T11^6 - 5*T11^5 + 54*T11^4 - 113*T11^3 + 1486*T11^2 - 3741*T11 + 16641 $$T_{13}^{6} + 2T_{13}^{5} + 44T_{13}^{4} - 16T_{13}^{3} + 1664T_{13}^{2} + 1280T_{13} + 1024$$ T13^6 + 2*T13^5 + 44*T13^4 - 16*T13^3 + 1664*T13^2 + 1280*T13 + 1024

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{6}$$
$3$ $$(T^{2} - T + 1)^{3}$$
$5$ $$(T^{2} + T + 1)^{3}$$
$7$ $$T^{6} + 4 T^{5} + 21 T^{4} + 4 T^{3} + \cdots + 144$$
$11$ $$T^{6} - 5 T^{5} + 54 T^{4} + \cdots + 16641$$
$13$ $$T^{6} + 2 T^{5} + 44 T^{4} + \cdots + 1024$$
$17$ $$T^{6} + 11 T^{5} + 91 T^{4} + \cdots + 144$$
$19$ $$T^{6} + 2 T^{5} + 64 T^{4} + \cdots + 20736$$
$23$ $$(T^{3} - 7 T^{2} + 6 T + 12)^{2}$$
$29$ $$(T^{3} - 12 T^{2} + 9 T + 146)^{2}$$
$31$ $$T^{6} - 8 T^{5} - 28 T^{4} + \cdots + 29791$$
$37$ $$T^{6} - 17 T^{5} + 203 T^{4} + \cdots + 15376$$
$41$ $$T^{6} + 12 T^{5} + 192 T^{4} + \cdots + 419904$$
$43$ $$T^{6} + 3 T^{5} + 45 T^{4} - 140 T^{3} + \cdots + 256$$
$47$ $$(T^{3} - 3 T^{2} - 90 T + 108)^{2}$$
$53$ $$T^{6} - 2 T^{5} + 59 T^{4} + \cdots + 8464$$
$59$ $$T^{6} + 12 T^{5} + 189 T^{4} + \cdots + 104976$$
$61$ $$(T^{3} + 8 T^{2} - 40 T - 248)^{2}$$
$67$ $$T^{6} + 17 T^{5} + 203 T^{4} + \cdots + 15376$$
$71$ $$T^{6} - 10 T^{5} + 108 T^{4} + \cdots + 9216$$
$73$ $$(T^{2} - 2 T + 4)^{3}$$
$79$ $$T^{6} + 3 T^{5} + 45 T^{4} - 140 T^{3} + \cdots + 256$$
$83$ $$T^{6} + 177 T^{4} - 424 T^{3} + \cdots + 44944$$
$89$ $$(T^{3} + 28 T^{2} + 220 T + 432)^{2}$$
$97$ $$(T^{3} - 20 T^{2} - 119 T + 2766)^{2}$$