# Properties

 Label 245.6.a.h Level $245$ Weight $6$ Character orbit 245.a Self dual yes Analytic conductor $39.294$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 245.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.2940358542$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - x^{5} - 109x^{4} + 41x^{3} + 2208x^{2} - 3204x + 560$$ x^6 - x^5 - 109*x^4 + 41*x^3 + 2208*x^2 - 3204*x + 560 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 35) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{3} - \beta_1 - 3) q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + 25 q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 5 \beta_1 - 16) q^{6} + (2 \beta_{5} + 4 \beta_{3} + 5 \beta_1 - 23) q^{8} + ( - 3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 65) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b3 - b1 - 3) * q^3 + (b2 - b1 + 5) * q^4 + 25 * q^5 + (-b5 + b4 - b3 + b2 - 5*b1 - 16) * q^6 + (2*b5 + 4*b3 + 5*b1 - 23) * q^8 + (-3*b5 - 2*b4 - 3*b3 - 4*b2 + 2*b1 + 65) * q^9 $$q + (\beta_1 - 1) q^{2} + (\beta_{3} - \beta_1 - 3) q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + 25 q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 5 \beta_1 - 16) q^{6} + (2 \beta_{5} + 4 \beta_{3} + 5 \beta_1 - 23) q^{8} + ( - 3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 65) q^{9} + (25 \beta_1 - 25) q^{10} + (\beta_{5} + 4 \beta_{4} - 9 \beta_{3} - 4 \beta_{2} - 32 \beta_1 - 151) q^{11} + (9 \beta_{5} - 5 \beta_{4} - 13 \beta_{3} - 10 \beta_{2} + 27 \beta_1 - 64) q^{12} + (3 \beta_{5} - 10 \beta_{4} - 7 \beta_{3} - 4 \beta_1 + 29) q^{13} + (25 \beta_{3} - 25 \beta_1 - 75) q^{15} + ( - 8 \beta_{5} + 8 \beta_{4} - 8 \beta_{3} - 11 \beta_{2} + 19 \beta_1 + 69) q^{16} + (4 \beta_{5} - 34 \beta_{3} - 20 \beta_{2} - 2 \beta_1 - 256) q^{17} + ( - 7 \beta_{5} - 5 \beta_{4} - 31 \beta_{3} - 20 \beta_{2} - 56 \beta_1 + 47) q^{18} + ( - 3 \beta_{5} + 8 \beta_{4} - 43 \beta_{3} + 8 \beta_{2} + 230 \beta_1 - 67) q^{19} + (25 \beta_{2} - 25 \beta_1 + 125) q^{20} + (15 \beta_{5} - 15 \beta_{4} + 39 \beta_{3} - 50 \beta_{2} - 308 \beta_1 - 1063) q^{22} + (53 \beta_{5} - 20 \beta_{4} - 34 \beta_{3} - 60 \beta_{2} + 49 \beta_1 - 646) q^{23} + ( - 13 \beta_{5} - 17 \beta_{4} - 73 \beta_{3} - 5 \beta_{2} - 47 \beta_1 + 1208) q^{24} + 625 q^{25} + ( - 39 \beta_{5} + 19 \beta_{4} - 119 \beta_{3} - 6 \beta_{2} + 210 \beta_1 - 495) q^{26} + (6 \beta_{5} + 60 \beta_{4} + 13 \beta_{3} + 60 \beta_{2} - 103 \beta_1 - 375) q^{27} + ( - 103 \beta_{5} + 38 \beta_{4} + 147 \beta_{3} + 112 \beta_{2} + \cdots + 240) q^{29}+ \cdots + (461 \beta_{5} + 524 \beta_{4} + 3201 \beta_{3} + 2532 \beta_{2} + \cdots - 32765) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b3 - b1 - 3) * q^3 + (b2 - b1 + 5) * q^4 + 25 * q^5 + (-b5 + b4 - b3 + b2 - 5*b1 - 16) * q^6 + (2*b5 + 4*b3 + 5*b1 - 23) * q^8 + (-3*b5 - 2*b4 - 3*b3 - 4*b2 + 2*b1 + 65) * q^9 + (25*b1 - 25) * q^10 + (b5 + 4*b4 - 9*b3 - 4*b2 - 32*b1 - 151) * q^11 + (9*b5 - 5*b4 - 13*b3 - 10*b2 + 27*b1 - 64) * q^12 + (3*b5 - 10*b4 - 7*b3 - 4*b1 + 29) * q^13 + (25*b3 - 25*b1 - 75) * q^15 + (-8*b5 + 8*b4 - 8*b3 - 11*b2 + 19*b1 + 69) * q^16 + (4*b5 - 34*b3 - 20*b2 - 2*b1 - 256) * q^17 + (-7*b5 - 5*b4 - 31*b3 - 20*b2 - 56*b1 + 47) * q^18 + (-3*b5 + 8*b4 - 43*b3 + 8*b2 + 230*b1 - 67) * q^19 + (25*b2 - 25*b1 + 125) * q^20 + (15*b5 - 15*b4 + 39*b3 - 50*b2 - 308*b1 - 1063) * q^22 + (53*b5 - 20*b4 - 34*b3 - 60*b2 + 49*b1 - 646) * q^23 + (-13*b5 - 17*b4 - 73*b3 - 5*b2 - 47*b1 + 1208) * q^24 + 625 * q^25 + (-39*b5 + 19*b4 - 119*b3 - 6*b2 + 210*b1 - 495) * q^26 + (6*b5 + 60*b4 + 13*b3 + 60*b2 - 103*b1 - 375) * q^27 + (-103*b5 + 38*b4 + 147*b3 + 112*b2 - 264*b1 + 240) * q^29 + (-25*b5 + 25*b4 - 25*b3 + 25*b2 - 125*b1 - 400) * q^30 + (-18*b5 + 68*b4 - 148*b3 + 12*b2 - 434*b1 + 1206) * q^31 + (-30*b5 - 40*b4 - 52*b3 - 40*b2 - 611*b1 + 1649) * q^32 + (-40*b5 - 40*b4 - 192*b3 + 704*b1 - 866) * q^33 + (-14*b5 - 26*b4 - 54*b3 - 74*b2 - 792*b1 - 198) * q^34 + (81*b5 + 29*b4 + b3 - 38*b2 - 588*b1 - 4313) * q^36 + (25*b5 + 10*b4 + 9*b3 + 220*b2 + 14*b1 - 2445) * q^37 + (97*b5 - 65*b4 + 177*b3 + 140*b2 + 136*b1 + 7679) * q^38 + (244*b5 + 36*b4 + 174*b3 + 100*b2 - 994*b1 - 2460) * q^39 + (50*b5 + 100*b3 + 125*b1 - 575) * q^40 + (62*b5 - 112*b4 - 198*b3 - 168*b2 - 44*b1 - 607) * q^41 + (112*b5 - 120*b4 - 193*b3 - 40*b2 - 871*b1 - 3909) * q^43 + (-261*b5 - 29*b4 - 161*b3 - 92*b2 - 1372*b1 - 4355) * q^44 + (-75*b5 - 50*b4 - 75*b3 - 100*b2 + 50*b1 + 1625) * q^45 + (-272*b5 + 112*b4 - 552*b3 + 133*b2 - 1741*b1 + 1279) * q^46 + (135*b5 + 80*b4 + 59*b3 + 120*b2 + 1402*b1 - 7483) * q^47 + (-267*b5 + 95*b4 + 291*b3 + 70*b2 + 451*b1 - 1988) * q^48 + (625*b1 - 625) * q^50 + (-6*b5 + 76*b4 + 4*b3 + 284*b2 - 38*b1 - 7126) * q^51 + (165*b5 + 85*b4 + 625*b3 - 190*b2 - 938*b1 + 6273) * q^52 + (-155*b5 - 110*b4 + 567*b3 + 240*b2 - 1068*b1 - 135) * q^53 + (335*b5 - 95*b4 + 935*b3 + 7*b2 + 733*b1 - 3238) * q^54 + (25*b5 + 100*b4 - 225*b3 - 100*b2 - 800*b1 - 3775) * q^55 + (-166*b5 + 220*b4 - 312*b3 + 220*b2 + 30*b1 - 16396) * q^57 + (435*b5 - 135*b4 + 963*b3 - 270*b2 + 2071*b1 - 6118) * q^58 + (-598*b5 + 68*b4 + 342*b3 + 476*b2 - 968*b1 + 10892) * q^59 + (225*b5 - 125*b4 - 325*b3 - 250*b2 + 675*b1 - 1600) * q^60 + (-443*b5 - 22*b4 + 727*b3 + 112*b2 - 2444*b1 - 11694) * q^61 + (480*b5 - 320*b4 + 1048*b3 - 790*b2 + 772*b1 - 18184) * q^62 + (128*b5 - 288*b4 - 272*b3 - 523*b2 + 155*b1 - 26107) * q^64 + (75*b5 - 250*b4 - 175*b3 - 100*b1 + 725) * q^65 + (112*b5 - 192*b4 - 208*b3 + 160*b2 - 282*b1 + 23226) * q^66 + (-234*b5 + 260*b4 + 1901*b3 - 460*b2 - 239*b1 - 23351) * q^67 + (-298*b5 - 30*b4 + 562*b3 - 390*b2 - 2156*b1 - 20074) * q^68 + (219*b5 - 34*b4 - 451*b3 + 904*b2 - 3328*b1 - 2732) * q^69 + (-114*b5 - 116*b4 - 1874*b3 + 396*b2 + 3068*b1 - 26588) * q^71 + (101*b5 + 265*b4 + 1025*b3 + 340*b2 - 3416*b1 - 19105) * q^72 + (1010*b5 + 20*b4 + 276*b3 - 500*b2 + 34*b1 + 882) * q^73 + (421*b5 + 39*b4 + 941*b3 + 352*b2 + 4662*b1 - 363) * q^74 + (625*b3 - 625*b1 - 1875) * q^75 + (-255*b5 + 245*b4 + 785*b3 + 762*b2 + 6228*b1 - 2537) * q^76 + (-318*b5 + 590*b4 + 170*b3 + 430*b2 + 2084*b1 - 36386) * q^78 + (-60*b5 + 380*b4 - 820*b3 - 1116*b2 + 2576*b1 - 37458) * q^79 + (-200*b5 + 200*b4 - 200*b3 - 275*b2 + 475*b1 + 1725) * q^80 + (84*b5 - 284*b4 - 2196*b3 - 184*b2 + 8060*b1 - 8851) * q^81 + (-710*b5 + 150*b4 - 1942*b3 - 360*b2 - 3461*b1 - 4861) * q^82 + (-222*b5 + 540*b4 - 1275*b3 + 1420*b2 + 1645*b1 - 31123) * q^83 + (100*b5 - 850*b3 - 500*b2 - 50*b1 - 6400) * q^85 + (-591*b5 + 271*b4 - 1631*b3 - 849*b2 - 2115*b1 - 34200) * q^86 + (-600*b5 - 160*b4 - 37*b3 - 2120*b2 + 7721*b1 + 34477) * q^87 + (-97*b5 - 145*b4 - 1281*b3 - 1230*b2 + 936*b1 - 7395) * q^88 + (429*b5 + 186*b4 - 3531*b3 - 140*b2 - 6634*b1 - 6542) * q^89 + (-175*b5 - 125*b4 - 775*b3 - 500*b2 - 1400*b1 + 1175) * q^90 + (114*b5 - 680*b4 + 4060*b3 - 1880*b2 + 1055*b1 - 47249) * q^92 + (-310*b5 - 580*b4 - 56*b3 - 660*b2 + 9790*b1 - 34166) * q^93 + (231*b5 + 169*b4 + 1111*b3 + 2180*b2 - 3666*b1 + 55563) * q^94 + (-75*b5 + 200*b4 - 1075*b3 + 200*b2 + 5750*b1 - 1675) * q^95 + (1179*b5 + 111*b4 + 3999*b3 + 195*b2 - 2559*b1 - 9528) * q^96 + (-1744*b5 - 120*b4 - 464*b3 - 1480*b2 + 2152*b1 + 12738) * q^97 + (461*b5 + 524*b4 + 3201*b3 + 2532*b2 + 1922*b1 - 32765) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 5 q^{2} - 20 q^{3} + 31 q^{4} + 150 q^{5} - 96 q^{6} - 135 q^{8} + 378 q^{9}+O(q^{10})$$ 6 * q - 5 * q^2 - 20 * q^3 + 31 * q^4 + 150 * q^5 - 96 * q^6 - 135 * q^8 + 378 * q^9 $$6 q - 5 q^{2} - 20 q^{3} + 31 q^{4} + 150 q^{5} - 96 q^{6} - 135 q^{8} + 378 q^{9} - 125 q^{10} - 924 q^{11} - 370 q^{12} + 150 q^{13} - 500 q^{15} + 435 q^{16} - 1540 q^{17} + 195 q^{18} - 92 q^{19} + 775 q^{20} - 6855 q^{22} - 3920 q^{23} + 7200 q^{24} + 3750 q^{25} - 2635 q^{26} - 2060 q^{27} + 1264 q^{29} - 2400 q^{30} + 7160 q^{31} + 9105 q^{32} - 4460 q^{33} - 2166 q^{34} - 26375 q^{36} - 14170 q^{37} + 46215 q^{38} - 15376 q^{39} - 3375 q^{40} - 4098 q^{41} - 24460 q^{43} - 27873 q^{44} + 9450 q^{45} + 6815 q^{46} - 42940 q^{47} - 11610 q^{48} - 3125 q^{50} - 42008 q^{51} + 36115 q^{52} - 2450 q^{53} - 19566 q^{54} - 23100 q^{55} - 97100 q^{57} - 36110 q^{58} + 64600 q^{59} - 9250 q^{60} - 73620 q^{61} - 111440 q^{62} - 157997 q^{64} + 3750 q^{65} + 139138 q^{66} - 142620 q^{67} - 124330 q^{68} - 17344 q^{69} - 154256 q^{71} - 117495 q^{72} + 5120 q^{73} + 2785 q^{74} - 12500 q^{75} - 7775 q^{76} - 214090 q^{78} - 222504 q^{79} + 10875 q^{80} - 43986 q^{81} - 31665 q^{82} - 179580 q^{83} - 38500 q^{85} - 207160 q^{86} + 209300 q^{87} - 45145 q^{88} - 41648 q^{89} + 4875 q^{90} - 292185 q^{92} - 198520 q^{93} + 333699 q^{94} - 2300 q^{95} - 61824 q^{96} + 73980 q^{97} - 190772 q^{99}+O(q^{100})$$ 6 * q - 5 * q^2 - 20 * q^3 + 31 * q^4 + 150 * q^5 - 96 * q^6 - 135 * q^8 + 378 * q^9 - 125 * q^10 - 924 * q^11 - 370 * q^12 + 150 * q^13 - 500 * q^15 + 435 * q^16 - 1540 * q^17 + 195 * q^18 - 92 * q^19 + 775 * q^20 - 6855 * q^22 - 3920 * q^23 + 7200 * q^24 + 3750 * q^25 - 2635 * q^26 - 2060 * q^27 + 1264 * q^29 - 2400 * q^30 + 7160 * q^31 + 9105 * q^32 - 4460 * q^33 - 2166 * q^34 - 26375 * q^36 - 14170 * q^37 + 46215 * q^38 - 15376 * q^39 - 3375 * q^40 - 4098 * q^41 - 24460 * q^43 - 27873 * q^44 + 9450 * q^45 + 6815 * q^46 - 42940 * q^47 - 11610 * q^48 - 3125 * q^50 - 42008 * q^51 + 36115 * q^52 - 2450 * q^53 - 19566 * q^54 - 23100 * q^55 - 97100 * q^57 - 36110 * q^58 + 64600 * q^59 - 9250 * q^60 - 73620 * q^61 - 111440 * q^62 - 157997 * q^64 + 3750 * q^65 + 139138 * q^66 - 142620 * q^67 - 124330 * q^68 - 17344 * q^69 - 154256 * q^71 - 117495 * q^72 + 5120 * q^73 + 2785 * q^74 - 12500 * q^75 - 7775 * q^76 - 214090 * q^78 - 222504 * q^79 + 10875 * q^80 - 43986 * q^81 - 31665 * q^82 - 179580 * q^83 - 38500 * q^85 - 207160 * q^86 + 209300 * q^87 - 45145 * q^88 - 41648 * q^89 + 4875 * q^90 - 292185 * q^92 - 198520 * q^93 + 333699 * q^94 - 2300 * q^95 - 61824 * q^96 + 73980 * q^97 - 190772 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 109x^{4} + 41x^{3} + 2208x^{2} - 3204x + 560$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 36$$ v^2 - v - 36 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - 103\nu^{3} - 86\nu^{2} + 1744\nu - 704 ) / 48$$ (v^5 - 103*v^3 - 86*v^2 + 1744*v - 704) / 48 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + 6\nu^{4} + 103\nu^{3} - 460\nu^{2} - 2380\nu + 5552 ) / 48$$ (-v^5 + 6*v^4 + 103*v^3 - 460*v^2 - 2380*v + 5552) / 48 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + 115\nu^{3} + 50\nu^{2} - 2536\nu + 1736 ) / 24$$ (-v^5 + 115*v^3 + 50*v^2 - 2536*v + 1736) / 24
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 36$$ b2 + b1 + 36 $$\nu^{3}$$ $$=$$ $$2\beta_{5} + 4\beta_{3} + 3\beta_{2} + 69\beta _1 + 22$$ 2*b5 + 4*b3 + 3*b2 + 69*b1 + 22 $$\nu^{4}$$ $$=$$ $$8\beta_{4} + 8\beta_{3} + 91\beta_{2} + 197\beta _1 + 2468$$ 8*b4 + 8*b3 + 91*b2 + 197*b1 + 2468 $$\nu^{5}$$ $$=$$ $$206\beta_{5} + 460\beta_{3} + 395\beta_{2} + 5449\beta _1 + 6066$$ 206*b5 + 460*b3 + 395*b2 + 5449*b1 + 6066

## 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
 −8.17253 −6.22733 0.203319 1.35048 4.12050 9.72556
−9.17253 −14.3258 52.1353 25.0000 131.404 0 −184.691 −37.7723 −229.313
1.2 −7.22733 15.9209 20.2343 25.0000 −115.066 0 85.0347 10.4759 −180.683
1.3 −0.796681 −10.5748 −31.3653 25.0000 8.42477 0 50.4819 −131.173 −19.9170
1.4 0.350479 21.5910 −31.8772 25.0000 7.56720 0 −22.3876 223.173 8.76197
1.5 3.12050 −27.8717 −22.2625 25.0000 −86.9736 0 −169.326 533.832 78.0125
1.6 8.72556 −4.73965 44.1354 25.0000 −41.3561 0 105.888 −220.536 218.139
 $$n$$: 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
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 245.6.a.h 6
7.b odd 2 1 245.6.a.i 6
7.d odd 6 2 35.6.e.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.e.a 12 7.d odd 6 2
245.6.a.h 6 1.a even 1 1 trivial
245.6.a.i 6 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2}^{6} + 5T_{2}^{5} - 99T_{2}^{4} - 385T_{2}^{3} + 1682T_{2}^{2} + 900T_{2} - 504$$ T2^6 + 5*T2^5 - 99*T2^4 - 385*T2^3 + 1682*T2^2 + 900*T2 - 504 $$T_{3}^{6} + 20T_{3}^{5} - 718T_{3}^{4} - 13100T_{3}^{3} + 87921T_{3}^{2} + 2078280T_{3} + 6879276$$ T3^6 + 20*T3^5 - 718*T3^4 - 13100*T3^3 + 87921*T3^2 + 2078280*T3 + 6879276

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 5 T^{5} - 99 T^{4} - 385 T^{3} + \cdots - 504$$
$3$ $$T^{6} + 20 T^{5} - 718 T^{4} + \cdots + 6879276$$
$5$ $$(T - 25)^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6} + 924 T^{5} + \cdots + 23\!\cdots\!00$$
$13$ $$T^{6} - 150 T^{5} + \cdots + 37\!\cdots\!36$$
$17$ $$T^{6} + 1540 T^{5} + \cdots + 18\!\cdots\!04$$
$19$ $$T^{6} + 92 T^{5} + \cdots - 15\!\cdots\!96$$
$23$ $$T^{6} + 3920 T^{5} + \cdots - 17\!\cdots\!01$$
$29$ $$T^{6} - 1264 T^{5} + \cdots - 10\!\cdots\!00$$
$31$ $$T^{6} - 7160 T^{5} + \cdots + 60\!\cdots\!44$$
$37$ $$T^{6} + 14170 T^{5} + \cdots + 68\!\cdots\!84$$
$41$ $$T^{6} + 4098 T^{5} + \cdots + 10\!\cdots\!25$$
$43$ $$T^{6} + 24460 T^{5} + \cdots + 19\!\cdots\!56$$
$47$ $$T^{6} + 42940 T^{5} + \cdots + 31\!\cdots\!00$$
$53$ $$T^{6} + 2450 T^{5} + \cdots + 26\!\cdots\!16$$
$59$ $$T^{6} - 64600 T^{5} + \cdots + 36\!\cdots\!00$$
$61$ $$T^{6} + 73620 T^{5} + \cdots + 21\!\cdots\!04$$
$67$ $$T^{6} + 142620 T^{5} + \cdots - 18\!\cdots\!00$$
$71$ $$T^{6} + 154256 T^{5} + \cdots + 99\!\cdots\!24$$
$73$ $$T^{6} - 5120 T^{5} + \cdots - 51\!\cdots\!04$$
$79$ $$T^{6} + 222504 T^{5} + \cdots + 71\!\cdots\!36$$
$83$ $$T^{6} + 179580 T^{5} + \cdots + 35\!\cdots\!64$$
$89$ $$T^{6} + 41648 T^{5} + \cdots - 46\!\cdots\!56$$
$97$ $$T^{6} - 73980 T^{5} + \cdots + 99\!\cdots\!84$$