# Properties

 Label 405.4.a.l Level $405$ Weight $4$ Character orbit 405.a Self dual yes Analytic conductor $23.896$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 405.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$23.8957735523$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432$$ x^6 - 2*x^5 - 38*x^4 + 42*x^3 + 393*x^2 - 72*x - 432 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{3}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{3} - \beta_1 + 6) q^{4} + 5 q^{5} + ( - \beta_{4} - \beta_{2} - 2 \beta_1 + 7) q^{7} + ( - \beta_{4} + 2 \beta_{3} - 4 \beta_1 + 12) q^{8}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b3 - b1 + 6) * q^4 + 5 * q^5 + (-b4 - b2 - 2*b1 + 7) * q^7 + (-b4 + 2*b3 - 4*b1 + 12) * q^8 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{3} - \beta_1 + 6) q^{4} + 5 q^{5} + ( - \beta_{4} - \beta_{2} - 2 \beta_1 + 7) q^{7} + ( - \beta_{4} + 2 \beta_{3} - 4 \beta_1 + 12) q^{8} + ( - 5 \beta_1 + 5) q^{10} + ( - \beta_{5} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 14) q^{11} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 4) q^{13} + (3 \beta_{5} - \beta_{4} + 4 \beta_{3} - \beta_{2} - 7 \beta_1 + 32) q^{14} + (2 \beta_{5} - 3 \beta_{4} - 16 \beta_1 + 14) q^{16} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 19) q^{17} + ( - 2 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 7) q^{19} + (5 \beta_{3} - 5 \beta_1 + 30) q^{20} + (2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 10 \beta_{2} - 20 \beta_1 - 5) q^{22} + (6 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 34) q^{23} + 25 q^{25} + ( - 7 \beta_{5} + 6 \beta_{4} - 5 \beta_{3} + 15 \beta_{2} + 2 \beta_1 + 2) q^{26} + (3 \beta_{5} - 6 \beta_{4} + 13 \beta_{3} - 29 \beta_{2} - 40 \beta_1 + 70) q^{28} + ( - 3 \beta_{5} - 4 \beta_{4} + \beta_{3} + 8 \beta_{2} - 12 \beta_1 + 52) q^{29} + ( - 4 \beta_{5} - 3 \beta_{4} - 10 \beta_{3} + 6 \beta_{2} - 16 \beta_1 - 13) q^{31} + (6 \beta_{5} - \beta_{4} + 6 \beta_{3} - 24 \beta_{2} + 18 \beta_1 + 114) q^{32} + ( - 4 \beta_{5} + 9 \beta_{4} - 7 \beta_{3} + 38 \beta_{2} - 25 \beta_1 - 60) q^{34} + ( - 5 \beta_{4} - 5 \beta_{2} - 10 \beta_1 + 35) q^{35} + (3 \beta_{5} + 5 \beta_{4} - 5 \beta_{3} - 28 \beta_{2} - 10 \beta_1 - 29) q^{37} + ( - 4 \beta_{5} + 14 \beta_{4} - 14 \beta_{3} + 20 \beta_{2} + 31 \beta_1 - 9) q^{38} + ( - 5 \beta_{4} + 10 \beta_{3} - 20 \beta_1 + 60) q^{40} + (10 \beta_{5} - 2 \beta_{4} - 26 \beta_{3} - 7 \beta_{2} - 16 \beta_1 + 62) q^{41} + (\beta_{5} - \beta_{4} + 25 \beta_{3} + 38 \beta_{2} - 42 \beta_1 + 99) q^{43} + ( - 6 \beta_{5} - 3 \beta_{4} + 7 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 120) q^{44} + ( - 4 \beta_{5} - 13 \beta_{4} - 14 \beta_{3} - 74 \beta_{2} - 22 \beta_1 - 28) q^{46} + ( - 9 \beta_{5} - 6 \beta_{4} - 29 \beta_{3} + 14 \beta_{2} + 44 \beta_1 + 23) q^{47} + (12 \beta_{5} - 15 \beta_{4} + 16 \beta_{3} - 12 \beta_{2} - 26 \beta_1 + 15) q^{49} + ( - 25 \beta_1 + 25) q^{50} + ( - 19 \beta_{5} + 16 \beta_{4} - 11 \beta_{3} + 75 \beta_{2} + 28 \beta_1 - 82) q^{52} + (5 \beta_{5} - 24 \beta_{4} - 27 \beta_{3} - 58 \beta_{2} + 20 \beta_1 + 127) q^{53} + ( - 5 \beta_{5} + 5 \beta_{3} - 10 \beta_{2} + 10 \beta_1 + 70) q^{55} + (17 \beta_{5} - 20 \beta_{4} + 33 \beta_{3} - 57 \beta_{2} - 92 \beta_1 + 410) q^{56} + (4 \beta_{4} + 21 \beta_{3} + 44 \beta_{2} - 58 \beta_1 + 169) q^{58} + ( - 8 \beta_{4} + 6 \beta_{3} + 15 \beta_{2} + 30 \beta_1 + 294) q^{59} + (22 \beta_{5} + 2 \beta_{4} - 38 \beta_{3} + 30 \beta_{2} + 72 \beta_1 - 84) q^{61} + (19 \beta_{4} + 12 \beta_{3} + 54 \beta_{2} + 73 \beta_1 + 155) q^{62} + (10 \beta_{5} - \beta_{4} - 10 \beta_{3} - 96 \beta_{2} - 22 \beta_1 - 158) q^{64} + ( - 5 \beta_{5} + 10 \beta_{4} - 5 \beta_{3} + 15 \beta_{2} + 20) q^{65} + (10 \beta_{5} + 12 \beta_{4} - 30 \beta_{3} - 8 \beta_{2} + 164 \beta_1 - 36) q^{67} + ( - 32 \beta_{5} + 20 \beta_{4} - 8 \beta_{3} + 70 \beta_{2} + 54 \beta_1 + 28) q^{68} + (15 \beta_{5} - 5 \beta_{4} + 20 \beta_{3} - 5 \beta_{2} - 35 \beta_1 + 160) q^{70} + (9 \beta_{5} + 40 \beta_{4} - 9 \beta_{3} + 52 \beta_{2} - 82 \beta_1 + 268) q^{71} + ( - 7 \beta_{5} + 23 \beta_{4} + 33 \beta_{3} - 50 \beta_{2} + 138 \beta_1 + 101) q^{73} + (18 \beta_{5} + \beta_{4} - 5 \beta_{3} - 64 \beta_{2} + 59 \beta_1 + 200) q^{74} + ( - 32 \beta_{5} + 8 \beta_{4} - 41 \beta_{3} + 100 \beta_{2} + 77 \beta_1 - 374) q^{76} + ( - 9 \beta_{5} - 5 \beta_{4} + 19 \beta_{3} - 40 \beta_{2} + 46 \beta_1 + 123) q^{77} + ( - 12 \beta_{5} - 32 \beta_{4} + 12 \beta_{3} + 10 \beta_{2} + 144 \beta_1 - 172) q^{79} + (10 \beta_{5} - 15 \beta_{4} - 80 \beta_1 + 70) q^{80} + (11 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 127 \beta_{2} + 94 \beta_1 + 257) q^{82} + ( - 19 \beta_{5} + 27 \beta_{4} + 45 \beta_{3} + 34 \beta_{2} - 6 \beta_1 + 369) q^{83} + ( - 15 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} + 10 \beta_{2} + 30 \beta_1 + 95) q^{85} + ( - 36 \beta_{5} - 29 \beta_{4} + 69 \beta_{3} + 26 \beta_{2} - 249 \beta_1 + 552) q^{86} + ( - 12 \beta_{5} - 8 \beta_{4} + 26 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 214) q^{88} + (21 \beta_{5} + 18 \beta_{4} + 33 \beta_{3} - 76 \beta_{2} + 96 \beta_1 + 474) q^{89} + ( - 18 \beta_{5} + 21 \beta_{4} - 34 \beta_{3} + 44 \beta_{2} + 106 \beta_1 - 538) q^{91} + (52 \beta_{5} - 11 \beta_{4} + 50 \beta_{3} - 10 \beta_{2} + 64 \beta_1 + 142) q^{92} + ( - 2 \beta_{5} + 50 \beta_{4} - 61 \beta_{3} + 122 \beta_{2} + 151 \beta_1 - 644) q^{94} + ( - 10 \beta_{5} + 20 \beta_{4} - 20 \beta_{3} - 20 \beta_{2} + 10 \beta_1 - 35) q^{95} + ( - 41 \beta_{5} - 39 \beta_{4} - 13 \beta_{3} + 72 \beta_{2} + 30 \beta_1 - 125) q^{97} + (42 \beta_{5} - 67 \beta_{4} + 72 \beta_{3} - 156 \beta_{2} - 111 \beta_1 + 345) q^{98}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b3 - b1 + 6) * q^4 + 5 * q^5 + (-b4 - b2 - 2*b1 + 7) * q^7 + (-b4 + 2*b3 - 4*b1 + 12) * q^8 + (-5*b1 + 5) * q^10 + (-b5 + b3 - 2*b2 + 2*b1 + 14) * q^11 + (-b5 + 2*b4 - b3 + 3*b2 + 4) * q^13 + (3*b5 - b4 + 4*b3 - b2 - 7*b1 + 32) * q^14 + (2*b5 - 3*b4 - 16*b1 + 14) * q^16 + (-3*b5 + b4 + b3 + 2*b2 + 6*b1 + 19) * q^17 + (-2*b5 + 4*b4 - 4*b3 - 4*b2 + 2*b1 - 7) * q^19 + (5*b3 - 5*b1 + 30) * q^20 + (2*b5 + 2*b4 - b3 + 10*b2 - 20*b1 - 5) * q^22 + (6*b5 + 3*b4 - 2*b3 - 2*b2 + 6*b1 + 34) * q^23 + 25 * q^25 + (-7*b5 + 6*b4 - 5*b3 + 15*b2 + 2*b1 + 2) * q^26 + (3*b5 - 6*b4 + 13*b3 - 29*b2 - 40*b1 + 70) * q^28 + (-3*b5 - 4*b4 + b3 + 8*b2 - 12*b1 + 52) * q^29 + (-4*b5 - 3*b4 - 10*b3 + 6*b2 - 16*b1 - 13) * q^31 + (6*b5 - b4 + 6*b3 - 24*b2 + 18*b1 + 114) * q^32 + (-4*b5 + 9*b4 - 7*b3 + 38*b2 - 25*b1 - 60) * q^34 + (-5*b4 - 5*b2 - 10*b1 + 35) * q^35 + (3*b5 + 5*b4 - 5*b3 - 28*b2 - 10*b1 - 29) * q^37 + (-4*b5 + 14*b4 - 14*b3 + 20*b2 + 31*b1 - 9) * q^38 + (-5*b4 + 10*b3 - 20*b1 + 60) * q^40 + (10*b5 - 2*b4 - 26*b3 - 7*b2 - 16*b1 + 62) * q^41 + (b5 - b4 + 25*b3 + 38*b2 - 42*b1 + 99) * q^43 + (-6*b5 - 3*b4 + 7*b3 + 2*b2 - 5*b1 + 120) * q^44 + (-4*b5 - 13*b4 - 14*b3 - 74*b2 - 22*b1 - 28) * q^46 + (-9*b5 - 6*b4 - 29*b3 + 14*b2 + 44*b1 + 23) * q^47 + (12*b5 - 15*b4 + 16*b3 - 12*b2 - 26*b1 + 15) * q^49 + (-25*b1 + 25) * q^50 + (-19*b5 + 16*b4 - 11*b3 + 75*b2 + 28*b1 - 82) * q^52 + (5*b5 - 24*b4 - 27*b3 - 58*b2 + 20*b1 + 127) * q^53 + (-5*b5 + 5*b3 - 10*b2 + 10*b1 + 70) * q^55 + (17*b5 - 20*b4 + 33*b3 - 57*b2 - 92*b1 + 410) * q^56 + (4*b4 + 21*b3 + 44*b2 - 58*b1 + 169) * q^58 + (-8*b4 + 6*b3 + 15*b2 + 30*b1 + 294) * q^59 + (22*b5 + 2*b4 - 38*b3 + 30*b2 + 72*b1 - 84) * q^61 + (19*b4 + 12*b3 + 54*b2 + 73*b1 + 155) * q^62 + (10*b5 - b4 - 10*b3 - 96*b2 - 22*b1 - 158) * q^64 + (-5*b5 + 10*b4 - 5*b3 + 15*b2 + 20) * q^65 + (10*b5 + 12*b4 - 30*b3 - 8*b2 + 164*b1 - 36) * q^67 + (-32*b5 + 20*b4 - 8*b3 + 70*b2 + 54*b1 + 28) * q^68 + (15*b5 - 5*b4 + 20*b3 - 5*b2 - 35*b1 + 160) * q^70 + (9*b5 + 40*b4 - 9*b3 + 52*b2 - 82*b1 + 268) * q^71 + (-7*b5 + 23*b4 + 33*b3 - 50*b2 + 138*b1 + 101) * q^73 + (18*b5 + b4 - 5*b3 - 64*b2 + 59*b1 + 200) * q^74 + (-32*b5 + 8*b4 - 41*b3 + 100*b2 + 77*b1 - 374) * q^76 + (-9*b5 - 5*b4 + 19*b3 - 40*b2 + 46*b1 + 123) * q^77 + (-12*b5 - 32*b4 + 12*b3 + 10*b2 + 144*b1 - 172) * q^79 + (10*b5 - 15*b4 - 80*b1 + 70) * q^80 + (11*b5 - 6*b4 - 6*b3 - 127*b2 + 94*b1 + 257) * q^82 + (-19*b5 + 27*b4 + 45*b3 + 34*b2 - 6*b1 + 369) * q^83 + (-15*b5 + 5*b4 + 5*b3 + 10*b2 + 30*b1 + 95) * q^85 + (-36*b5 - 29*b4 + 69*b3 + 26*b2 - 249*b1 + 552) * q^86 + (-12*b5 - 8*b4 + 26*b3 - 6*b2 - 2*b1 + 214) * q^88 + (21*b5 + 18*b4 + 33*b3 - 76*b2 + 96*b1 + 474) * q^89 + (-18*b5 + 21*b4 - 34*b3 + 44*b2 + 106*b1 - 538) * q^91 + (52*b5 - 11*b4 + 50*b3 - 10*b2 + 64*b1 + 142) * q^92 + (-2*b5 + 50*b4 - 61*b3 + 122*b2 + 151*b1 - 644) * q^94 + (-10*b5 + 20*b4 - 20*b3 - 20*b2 + 10*b1 - 35) * q^95 + (-41*b5 - 39*b4 - 13*b3 + 72*b2 + 30*b1 - 125) * q^97 + (42*b5 - 67*b4 + 72*b3 - 156*b2 - 111*b1 + 345) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 4 q^{2} + 34 q^{4} + 30 q^{5} + 40 q^{7} + 66 q^{8}+O(q^{10})$$ 6 * q + 4 * q^2 + 34 * q^4 + 30 * q^5 + 40 * q^7 + 66 * q^8 $$6 q + 4 q^{2} + 34 q^{4} + 30 q^{5} + 40 q^{7} + 66 q^{8} + 20 q^{10} + 88 q^{11} + 20 q^{13} + 180 q^{14} + 58 q^{16} + 124 q^{17} - 46 q^{19} + 170 q^{20} - 74 q^{22} + 210 q^{23} + 150 q^{25} + 4 q^{26} + 352 q^{28} + 296 q^{29} - 104 q^{31} + 722 q^{32} - 428 q^{34} + 200 q^{35} - 204 q^{37} - 20 q^{38} + 330 q^{40} + 344 q^{41} + 512 q^{43} + 716 q^{44} - 186 q^{46} + 238 q^{47} + 68 q^{49} + 100 q^{50} - 468 q^{52} + 850 q^{53} + 440 q^{55} + 2316 q^{56} + 890 q^{58} + 1840 q^{59} - 364 q^{61} + 1038 q^{62} - 990 q^{64} + 100 q^{65} + 88 q^{67} + 236 q^{68} + 900 q^{70} + 1364 q^{71} + 836 q^{73} + 1316 q^{74} - 2106 q^{76} + 840 q^{77} - 680 q^{79} + 290 q^{80} + 1742 q^{82} + 2148 q^{83} + 620 q^{85} + 2872 q^{86} + 1296 q^{88} + 3000 q^{89} - 3058 q^{91} + 1002 q^{92} - 3662 q^{94} - 230 q^{95} - 612 q^{97} + 1982 q^{98}+O(q^{100})$$ 6 * q + 4 * q^2 + 34 * q^4 + 30 * q^5 + 40 * q^7 + 66 * q^8 + 20 * q^10 + 88 * q^11 + 20 * q^13 + 180 * q^14 + 58 * q^16 + 124 * q^17 - 46 * q^19 + 170 * q^20 - 74 * q^22 + 210 * q^23 + 150 * q^25 + 4 * q^26 + 352 * q^28 + 296 * q^29 - 104 * q^31 + 722 * q^32 - 428 * q^34 + 200 * q^35 - 204 * q^37 - 20 * q^38 + 330 * q^40 + 344 * q^41 + 512 * q^43 + 716 * q^44 - 186 * q^46 + 238 * q^47 + 68 * q^49 + 100 * q^50 - 468 * q^52 + 850 * q^53 + 440 * q^55 + 2316 * q^56 + 890 * q^58 + 1840 * q^59 - 364 * q^61 + 1038 * q^62 - 990 * q^64 + 100 * q^65 + 88 * q^67 + 236 * q^68 + 900 * q^70 + 1364 * q^71 + 836 * q^73 + 1316 * q^74 - 2106 * q^76 + 840 * q^77 - 680 * q^79 + 290 * q^80 + 1742 * q^82 + 2148 * q^83 + 620 * q^85 + 2872 * q^86 + 1296 * q^88 + 3000 * q^89 - 3058 * q^91 + 1002 * q^92 - 3662 * q^94 - 230 * q^95 - 612 * q^97 + 1982 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - 2\nu^{4} - 26\nu^{3} + 30\nu^{2} + 141\nu - 36 ) / 24$$ (v^5 - 2*v^4 - 26*v^3 + 30*v^2 + 141*v - 36) / 24 $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 13$$ v^2 - v - 13 $$\beta_{4}$$ $$=$$ $$\nu^{3} - \nu^{2} - 19\nu + 1$$ v^3 - v^2 - 19*v + 1 $$\beta_{5}$$ $$=$$ $$( \nu^{4} - \nu^{3} - 21\nu^{2} + 3\nu + 30 ) / 2$$ (v^4 - v^3 - 21*v^2 + 3*v + 30) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta _1 + 13$$ b3 + b1 + 13 $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + 20\beta _1 + 12$$ b4 + b3 + 20*b1 + 12 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + \beta_{4} + 22\beta_{3} + 38\beta _1 + 255$$ 2*b5 + b4 + 22*b3 + 38*b1 + 255 $$\nu^{5}$$ $$=$$ $$4\beta_{5} + 28\beta_{4} + 40\beta_{3} + 24\beta_{2} + 425\beta _1 + 468$$ 4*b5 + 28*b4 + 40*b3 + 24*b2 + 425*b1 + 468

## 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
 5.11734 4.57457 1.14915 −1.07326 −3.53444 −4.23336
−4.11734 0 8.95250 5.00000 0 −20.0229 −3.92177 0 −20.5867
1.2 −3.57457 0 4.77759 5.00000 0 14.1597 11.5188 0 −17.8729
1.3 −0.149150 0 −7.97775 5.00000 0 20.1424 2.38308 0 −0.745751
1.4 2.07326 0 −3.70159 5.00000 0 −4.66112 −24.2604 0 10.3663
1.5 4.53444 0 12.5612 5.00000 0 −2.63618 20.6823 0 22.6722
1.6 5.23336 0 19.3881 5.00000 0 33.0180 59.5981 0 26.1668
 $$n$$: 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
$$3$$ $$-1$$
$$5$$ $$-1$$

## 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 405.4.a.l yes 6
3.b odd 2 1 405.4.a.k 6
5.b even 2 1 2025.4.a.y 6
9.c even 3 2 405.4.e.w 12
9.d odd 6 2 405.4.e.x 12
15.d odd 2 1 2025.4.a.z 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.k 6 3.b odd 2 1
405.4.a.l yes 6 1.a even 1 1 trivial
405.4.e.w 12 9.c even 3 2
405.4.e.x 12 9.d odd 6 2
2025.4.a.y 6 5.b even 2 1
2025.4.a.z 6 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 4T_{2}^{5} - 33T_{2}^{4} + 110T_{2}^{3} + 286T_{2}^{2} - 684T_{2} - 108$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(405))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 4 T^{5} - 33 T^{4} + 110 T^{3} + \cdots - 108$$
$3$ $$T^{6}$$
$5$ $$(T - 5)^{6}$$
$7$ $$T^{6} - 40 T^{5} - 263 T^{4} + \cdots - 2316924$$
$11$ $$T^{6} - 88 T^{5} + 1518 T^{4} + \cdots + 1197108$$
$13$ $$T^{6} - 20 T^{5} + \cdots - 185793728$$
$17$ $$T^{6} - 124 T^{5} + \cdots + 105966288$$
$19$ $$T^{6} + 46 T^{5} + \cdots - 36821611175$$
$23$ $$T^{6} - 210 T^{5} + \cdots + 332569842768$$
$29$ $$T^{6} - 296 T^{5} + \cdots + 635447099088$$
$31$ $$T^{6} + 104 T^{5} + \cdots - 69859854216$$
$37$ $$T^{6} + 204 T^{5} + \cdots + 12008297128192$$
$41$ $$T^{6} - 344 T^{5} + \cdots - 86213802377403$$
$43$ $$T^{6} + \cdots - 180465347194400$$
$47$ $$T^{6} - 238 T^{5} + \cdots - 49451750433900$$
$53$ $$T^{6} - 850 T^{5} + \cdots + 15\!\cdots\!00$$
$59$ $$T^{6} + \cdots - 168320389359483$$
$61$ $$T^{6} + 364 T^{5} + \cdots - 20\!\cdots\!36$$
$67$ $$T^{6} - 88 T^{5} + \cdots + 19\!\cdots\!00$$
$71$ $$T^{6} - 1364 T^{5} + \cdots - 11\!\cdots\!84$$
$73$ $$T^{6} - 836 T^{5} + \cdots - 10\!\cdots\!36$$
$79$ $$T^{6} + 680 T^{5} + \cdots - 15\!\cdots\!72$$
$83$ $$T^{6} - 2148 T^{5} + \cdots + 41\!\cdots\!32$$
$89$ $$T^{6} - 3000 T^{5} + \cdots + 16\!\cdots\!48$$
$97$ $$T^{6} + 612 T^{5} + \cdots - 97\!\cdots\!24$$