Properties

 Label 2366.2.a.bf Level $2366$ Weight $2$ Character orbit 2366.a Self dual yes Analytic conductor $18.893$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2366 = 2 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2366.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$18.8926051182$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.285686784.1 Defining polynomial: $$x^{6} - 2x^{5} - 10x^{4} + 12x^{3} + 21x^{2} + 2x - 2$$ x^6 - 2*x^5 - 10*x^4 + 12*x^3 + 21*x^2 + 2*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 182) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 10x^{4} + 12x^{3} + 21x^{2} + 2x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 4\nu^{4} + 7\nu^{3} - 26\nu^{2} - 9\nu + 1 ) / 5$$ (-v^5 + 4*v^4 + 7*v^3 - 26*v^2 - 9*v + 1) / 5 $$\beta_{3}$$ $$=$$ $$( -2\nu^{5} + 3\nu^{4} + 24\nu^{3} - 17\nu^{2} - 68\nu - 13 ) / 5$$ (-2*v^5 + 3*v^4 + 24*v^3 - 17*v^2 - 68*v - 13) / 5 $$\beta_{4}$$ $$=$$ $$( 3\nu^{5} - 7\nu^{4} - 26\nu^{3} + 43\nu^{2} + 37\nu - 3 ) / 5$$ (3*v^5 - 7*v^4 - 26*v^3 + 43*v^2 + 37*v - 3) / 5 $$\beta_{5}$$ $$=$$ $$( 4\nu^{5} - 11\nu^{4} - 33\nu^{3} + 74\nu^{2} + 41\nu - 24 ) / 5$$ (4*v^5 - 11*v^4 - 33*v^3 + 74*v^2 + 41*v - 24) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 4$$ b5 - b4 + b2 + b1 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + \beta_{2} + 8\beta _1 + 3$$ b4 + b3 + b2 + 8*b1 + 3 $$\nu^{4}$$ $$=$$ $$7\beta_{5} - 5\beta_{4} + \beta_{3} + 11\beta_{2} + 13\beta _1 + 31$$ 7*b5 - 5*b4 + b3 + 11*b2 + 13*b1 + 31 $$\nu^{5}$$ $$=$$ $$2\beta_{5} + 13\beta_{4} + 11\beta_{3} + 20\beta_{2} + 73\beta _1 + 42$$ 2*b5 + 13*b4 + 11*b3 + 20*b2 + 73*b1 + 42

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.55629 −0.865515 −0.466545 0.252878 2.29079 3.34469
−1.00000 −2.55629 1.00000 3.48754 2.55629 −1.00000 −1.00000 3.53463 −3.48754
1.2 −1.00000 −0.865515 1.00000 3.71131 0.865515 −1.00000 −1.00000 −2.25088 −3.71131
1.3 −1.00000 −0.466545 1.00000 −3.38938 0.466545 −1.00000 −1.00000 −2.78234 3.38938
1.4 −1.00000 0.252878 1.00000 −1.14776 −0.252878 −1.00000 −1.00000 −2.93605 1.14776
1.5 −1.00000 2.29079 1.00000 0.901839 −2.29079 −1.00000 −1.00000 2.24770 −0.901839
1.6 −1.00000 3.34469 1.00000 −1.56356 −3.34469 −1.00000 −1.00000 8.18694 1.56356
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$1$$
$$13$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2366.2.a.bf 6
13.b even 2 1 2366.2.a.bh 6
13.d odd 4 2 2366.2.d.r 12
13.f odd 12 2 182.2.m.b 12
39.k even 12 2 1638.2.bj.g 12
52.l even 12 2 1456.2.cc.d 12
91.w even 12 2 1274.2.v.d 12
91.x odd 12 2 1274.2.o.d 12
91.ba even 12 2 1274.2.o.e 12
91.bc even 12 2 1274.2.m.c 12
91.bd odd 12 2 1274.2.v.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.m.b 12 13.f odd 12 2
1274.2.m.c 12 91.bc even 12 2
1274.2.o.d 12 91.x odd 12 2
1274.2.o.e 12 91.ba even 12 2
1274.2.v.d 12 91.w even 12 2
1274.2.v.e 12 91.bd odd 12 2
1456.2.cc.d 12 52.l even 12 2
1638.2.bj.g 12 39.k even 12 2
2366.2.a.bf 6 1.a even 1 1 trivial
2366.2.a.bh 6 13.b even 2 1
2366.2.d.r 12 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2366))$$:

 $$T_{3}^{6} - 2T_{3}^{5} - 10T_{3}^{4} + 12T_{3}^{3} + 21T_{3}^{2} + 2T_{3} - 2$$ T3^6 - 2*T3^5 - 10*T3^4 + 12*T3^3 + 21*T3^2 + 2*T3 - 2 $$T_{5}^{6} - 2T_{5}^{5} - 19T_{5}^{4} + 24T_{5}^{3} + 93T_{5}^{2} - 10T_{5} - 71$$ T5^6 - 2*T5^5 - 19*T5^4 + 24*T5^3 + 93*T5^2 - 10*T5 - 71

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{6}$$
$3$ $$T^{6} - 2 T^{5} - 10 T^{4} + 12 T^{3} + \cdots - 2$$
$5$ $$T^{6} - 2 T^{5} - 19 T^{4} + 24 T^{3} + \cdots - 71$$
$7$ $$(T + 1)^{6}$$
$11$ $$T^{6} + 2 T^{5} - 42 T^{4} - 96 T^{3} + \cdots - 704$$
$13$ $$T^{6}$$
$17$ $$T^{6} - 4 T^{5} - 39 T^{4} + 168 T^{3} + \cdots - 176$$
$19$ $$T^{6} - 4 T^{5} - 92 T^{4} + \cdots - 19232$$
$23$ $$T^{6} + 6 T^{5} - 68 T^{4} + \cdots + 3142$$
$29$ $$T^{6} - 10 T^{5} - 39 T^{4} + \cdots - 368$$
$31$ $$T^{6} - 2 T^{5} - 42 T^{4} - 48 T^{3} + \cdots - 32$$
$37$ $$T^{6} - 155 T^{4} + 8 T^{3} + 5752 T^{2} + \cdots - 32$$
$41$ $$T^{6} + 6 T^{5} - 95 T^{4} - 352 T^{3} + \cdots - 32$$
$43$ $$T^{6} - 26 T^{5} + 214 T^{4} + \cdots + 2944$$
$47$ $$T^{6} - 8 T^{5} - 104 T^{4} + \cdots + 5632$$
$53$ $$T^{6} - 18 T^{5} - 51 T^{4} + \cdots + 44928$$
$59$ $$T^{6} + 2 T^{5} - 82 T^{4} - 84 T^{3} + \cdots - 2$$
$61$ $$T^{6} - 28 T^{5} + 69 T^{4} + \cdots - 283487$$
$67$ $$T^{6} + 6 T^{5} - 122 T^{4} - 448 T^{3} + \cdots - 32$$
$71$ $$T^{6} + 4 T^{5} - 154 T^{4} + \cdots - 27392$$
$73$ $$T^{6} - 22 T^{5} - 63 T^{4} + \cdots + 52048$$
$79$ $$T^{6} - 22 T^{5} + 62 T^{4} + \cdots + 8032$$
$83$ $$T^{6} - 10 T^{5} - 182 T^{4} + \cdots + 29656$$
$89$ $$T^{6} - 4 T^{5} - 200 T^{4} + \cdots - 101504$$
$97$ $$T^{6} - 12 T^{5} - 224 T^{4} + \cdots - 80000$$