# Properties

 Label 418.2.e.i Level $418$ Weight $2$ Character orbit 418.e Analytic conductor $3.338$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.101617200.1 Defining polynomial: $$x^{6} + 8x^{4} - 4x^{3} + 64x^{2} - 16x + 4$$ x^6 + 8*x^4 - 4*x^3 + 64*x^2 - 16*x + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 8x^{4} - 4x^{3} + 64x^{2} - 16x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 2 ) / 8$$ (v^3 - 2) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{4} + 8\nu^{2} - 2\nu + 40 ) / 8$$ (v^4 + 8*v^2 - 2*v + 40) / 8 $$\beta_{4}$$ $$=$$ $$( \nu^{5} + 8\nu^{3} - 2\nu^{2} + 64\nu ) / 16$$ (v^5 + 8*v^3 - 2*v^2 + 64*v) / 16 $$\beta_{5}$$ $$=$$ $$( -5\nu^{5} - 40\nu^{3} + 26\nu^{2} - 320\nu + 80 ) / 16$$ (-5*v^5 - 40*v^3 + 26*v^2 - 320*v + 80) / 16
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 5\beta_{4} - 5$$ b5 + 5*b4 - 5 $$\nu^{3}$$ $$=$$ $$8\beta_{2} + 2$$ 8*b2 + 2 $$\nu^{4}$$ $$=$$ $$-8\beta_{5} - 40\beta_{4} + 8\beta_{3} + 2\beta_1$$ -8*b5 - 40*b4 + 8*b3 + 2*b1 $$\nu^{5}$$ $$=$$ $$2\beta_{5} + 26\beta_{4} - 64\beta_{2} - 64\beta _1 - 26$$ 2*b5 + 26*b4 - 64*b2 - 64*b1 - 26

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/418\mathbb{Z}\right)^\times$$.

 $$n$$ $$287$$ $$343$$ $$\chi(n)$$ $$-1 + \beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
45.1
 1.34700 + 2.33307i 0.126000 + 0.218239i −1.47300 − 2.55131i 1.34700 − 2.33307i 0.126000 − 0.218239i −1.47300 + 2.55131i
0.500000 + 0.866025i −1.34700 2.33307i −0.500000 + 0.866025i −1.00000 1.73205i 1.34700 2.33307i −0.693995 −1.00000 −2.12880 + 3.68720i 1.00000 1.73205i
45.2 0.500000 + 0.866025i −0.126000 0.218239i −0.500000 + 0.866025i −1.00000 1.73205i 0.126000 0.218239i 1.74800 −1.00000 1.46825 2.54308i 1.00000 1.73205i
45.3 0.500000 + 0.866025i 1.47300 + 2.55131i −0.500000 + 0.866025i −1.00000 1.73205i −1.47300 + 2.55131i 4.94600 −1.00000 −2.83944 + 4.91806i 1.00000 1.73205i
353.1 0.500000 0.866025i −1.34700 + 2.33307i −0.500000 0.866025i −1.00000 + 1.73205i 1.34700 + 2.33307i −0.693995 −1.00000 −2.12880 3.68720i 1.00000 + 1.73205i
353.2 0.500000 0.866025i −0.126000 + 0.218239i −0.500000 0.866025i −1.00000 + 1.73205i 0.126000 + 0.218239i 1.74800 −1.00000 1.46825 + 2.54308i 1.00000 + 1.73205i
353.3 0.500000 0.866025i 1.47300 2.55131i −0.500000 0.866025i −1.00000 + 1.73205i −1.47300 2.55131i 4.94600 −1.00000 −2.83944 4.91806i 1.00000 + 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 353.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.e.i 6
19.c even 3 1 inner 418.2.e.i 6
19.c even 3 1 7942.2.a.bd 3
19.d odd 6 1 7942.2.a.bj 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.i 6 1.a even 1 1 trivial
418.2.e.i 6 19.c even 3 1 inner
7942.2.a.bd 3 19.c even 3 1
7942.2.a.bj 3 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$:

 $$T_{3}^{6} + 8T_{3}^{4} + 4T_{3}^{3} + 64T_{3}^{2} + 16T_{3} + 4$$ T3^6 + 8*T3^4 + 4*T3^3 + 64*T3^2 + 16*T3 + 4 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{3}$$
$3$ $$T^{6} + 8 T^{4} + 4 T^{3} + 64 T^{2} + \cdots + 4$$
$5$ $$(T^{2} + 2 T + 4)^{3}$$
$7$ $$(T^{3} - 6 T^{2} + 4 T + 6)^{2}$$
$11$ $$(T - 1)^{6}$$
$13$ $$T^{6} + 7 T^{5} + 54 T^{4} + \cdots + 5041$$
$17$ $$T^{6} + 10 T^{5} + 90 T^{4} + \cdots + 2500$$
$19$ $$T^{6} + 5 T^{5} - 16 T^{4} + \cdots + 6859$$
$23$ $$T^{6} + 12 T^{5} + 128 T^{4} + \cdots + 6400$$
$29$ $$(T^{2} - T + 1)^{3}$$
$31$ $$(T^{3} - 2 T^{2} - 40 T + 116)^{2}$$
$37$ $$(T^{3} - 4 T^{2} - 70 T - 150)^{2}$$
$41$ $$T^{6} + 10 T^{5} + 102 T^{4} + \cdots + 2916$$
$43$ $$T^{6} + 5 T^{5} + 52 T^{4} + \cdots + 6561$$
$47$ $$T^{6} - 11 T^{5} + 162 T^{4} + \cdots + 251001$$
$53$ $$T^{6} + 128 T^{4} - 256 T^{3} + \cdots + 16384$$
$59$ $$T^{6}$$
$61$ $$T^{6} + 11 T^{5} + 146 T^{4} + \cdots + 140625$$
$67$ $$T^{6} + 72 T^{4} - 108 T^{3} + \cdots + 2916$$
$71$ $$T^{6} - 5 T^{5} + 52 T^{4} + \cdots + 6561$$
$73$ $$T^{6} + 8 T^{5} + 78 T^{4} + \cdots + 10404$$
$79$ $$T^{6} + 6 T^{5} + 96 T^{4} + \cdots + 36100$$
$83$ $$(T^{3} - 7 T^{2} - 95 T - 205)^{2}$$
$89$ $$T^{6} + T^{5} + 94 T^{4} - 179 T^{3} + \cdots + 1849$$
$97$ $$T^{6} - 7 T^{5} + 126 T^{4} + \cdots + 17161$$