# Properties

 Label 825.6.a.p Level $825$ Weight $6$ Character orbit 825.a Self dual yes Analytic conductor $132.317$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$132.316651346$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248$$ x^8 - x^7 - 176*x^6 + 272*x^5 + 9055*x^4 - 15851*x^3 - 118840*x^2 + 149572*x - 33248 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}\cdot 5$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{2} + 9 q^{3} + (\beta_{2} + \beta_1 + 13) q^{4} + ( - 9 \beta_1 - 9) q^{6} + ( - \beta_{6} - \beta_{2} - 3 \beta_1 + 9) q^{7} + ( - \beta_{3} - \beta_{2} - 8 \beta_1 - 25) q^{8} + 81 q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^2 + 9 * q^3 + (b2 + b1 + 13) * q^4 + (-9*b1 - 9) * q^6 + (-b6 - b2 - 3*b1 + 9) * q^7 + (-b3 - b2 - 8*b1 - 25) * q^8 + 81 * q^9 $$q + ( - \beta_1 - 1) q^{2} + 9 q^{3} + (\beta_{2} + \beta_1 + 13) q^{4} + ( - 9 \beta_1 - 9) q^{6} + ( - \beta_{6} - \beta_{2} - 3 \beta_1 + 9) q^{7} + ( - \beta_{3} - \beta_{2} - 8 \beta_1 - 25) q^{8} + 81 q^{9} - 121 q^{11} + (9 \beta_{2} + 9 \beta_1 + 117) q^{12} + (\beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{3} - 3 \beta_{2} + 42 \beta_1 - 52) q^{13} + (\beta_{7} + 5 \beta_{6} - 2 \beta_{5} - \beta_{3} + 5 \beta_{2} + 7 \beta_1 + 128) q^{14} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 7 \beta_{2} + 31 \beta_1 - 43) q^{16} + (10 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 5 \beta_{2} - 6 \beta_1 - 35) q^{17} + ( - 81 \beta_1 - 81) q^{18} + ( - 3 \beta_{7} - 7 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \cdots - 125) q^{19}+ \cdots - 9801 q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^2 + 9 * q^3 + (b2 + b1 + 13) * q^4 + (-9*b1 - 9) * q^6 + (-b6 - b2 - 3*b1 + 9) * q^7 + (-b3 - b2 - 8*b1 - 25) * q^8 + 81 * q^9 - 121 * q^11 + (9*b2 + 9*b1 + 117) * q^12 + (b7 - b6 + b5 + 2*b3 - 3*b2 + 42*b1 - 52) * q^13 + (b7 + 5*b6 - 2*b5 - b3 + 5*b2 + 7*b1 + 128) * q^14 + (-2*b6 + 2*b5 + 2*b4 + 2*b3 - 7*b2 + 31*b1 - 43) * q^16 + (10*b6 - b5 - b4 - b3 - 5*b2 - 6*b1 - 35) * q^17 + (-81*b1 - 81) * q^18 + (-3*b7 - 7*b6 + 4*b5 - 3*b4 + 3*b3 + 2*b2 + 34*b1 - 125) * q^19 + (-9*b6 - 9*b2 - 27*b1 + 81) * q^21 + (121*b1 + 121) * q^22 + (-6*b7 + b6 - 9*b5 + 3*b4 + 7*b3 - 6*b2 - b1 - 739) * q^23 + (-9*b3 - 9*b2 - 72*b1 - 225) * q^24 + (-b7 + 11*b6 - 6*b5 - 8*b4 - 7*b3 - 76*b2 + 100*b1 - 1815) * q^26 + 729 * q^27 + (-13*b7 - 11*b6 + 18*b5 + 4*b4 + 9*b3 + 24*b2 - 137*b1 - 728) * q^28 + (-b7 - 6*b6 - 22*b5 + 7*b4 + 5*b3 - 15*b2 + 101*b1 + 126) * q^29 + (9*b7 + 21*b6 + 9*b5 - 22*b3 - 37*b2 + 44*b1 - 598) * q^31 + (-4*b7 + 8*b6 - 20*b5 + 13*b3 - 11*b2 + 474*b1 - 529) * q^32 - 1089 * q^33 + (-7*b7 - 49*b6 + 28*b5 + 36*b3 - 6*b2 + 276*b1 + 258) * q^34 + (81*b2 + 81*b1 + 1053) * q^36 + (b7 + 45*b6 + 41*b5 + 10*b4 + 20*b3 + 141*b2 + 426*b1 + 2049) * q^37 + (42*b7 + 104*b6 - 22*b5 - 20*b4 - b3 - 72*b2 + 36*b1 - 1379) * q^38 + (9*b7 - 9*b6 + 9*b5 + 18*b3 - 27*b2 + 378*b1 - 468) * q^39 + (-26*b7 + 58*b6 + 21*b5 + 3*b4 + 15*b3 - 93*b2 - 116*b1 - 1613) * q^41 + (9*b7 + 45*b6 - 18*b5 - 9*b3 + 45*b2 + 63*b1 + 1152) * q^42 + (21*b7 - 5*b6 - 35*b5 - 38*b4 - 121*b2 - 56*b1 - 2806) * q^43 + (-121*b2 - 121*b1 - 1573) * q^44 + (-10*b7 - 48*b6 - 18*b5 + 28*b4 + 25*b3 - 93*b2 + 808*b1 + 1073) * q^46 + (17*b7 + b6 - 23*b5 - 2*b4 - 88*b3 + 325*b2 + 490*b1 - 1013) * q^47 + (-18*b6 + 18*b5 + 18*b4 + 18*b3 - 63*b2 + 279*b1 - 387) * q^48 + (47*b7 + 10*b6 - 22*b5 + b4 - 29*b3 - 89*b2 + 995*b1 - 5560) * q^49 + (90*b6 - 9*b5 - 9*b4 - 9*b3 - 45*b2 - 54*b1 - 315) * q^51 + (-11*b7 + 3*b6 + 46*b5 - 4*b4 + 118*b3 + 29*b2 + 2637*b1 - 925) * q^52 + (-38*b7 - 4*b6 + 106*b5 + 48*b4 + 60*b3 + 128*b2 + 438*b1 - 872) * q^53 + (-729*b1 - 729) * q^54 + (47*b7 + 37*b6 - 54*b5 - 12*b4 - 66*b3 - 38*b2 + 20*b1 + 2602) * q^56 + (-27*b7 - 63*b6 + 36*b5 - 27*b4 + 27*b3 + 18*b2 + 306*b1 - 1125) * q^57 + (-69*b7 - 151*b6 - 8*b5 + 64*b4 + 18*b3 - 165*b2 - 73*b1 - 4047) * q^58 + (-27*b7 + 147*b6 + 77*b5 + 22*b4 - 68*b3 - 21*b2 - 620*b1 - 9673) * q^59 + (64*b7 + 9*b6 - 115*b5 - 23*b4 + 17*b3 - 796*b2 + 303*b1 + 1624) * q^61 + (-39*b7 - 131*b6 + 86*b5 + 8*b4 + 47*b3 + 270*b2 + 2070*b1 - 1819) * q^62 + (-81*b6 - 81*b2 - 243*b1 + 729) * q^63 + (-32*b7 - 54*b6 - 42*b5 - 42*b4 + 22*b3 - 511*b2 - 413*b1 - 18507) * q^64 + (1089*b1 + 1089) * q^66 + (40*b7 - 67*b6 - 203*b5 + b4 - 199*b3 + 472*b2 + 1531*b1 - 2952) * q^67 + (133*b7 + 193*b6 - 208*b5 - 82*b4 - 159*b3 - 546*b2 - 419*b1 - 11334) * q^68 + (-54*b7 + 9*b6 - 81*b5 + 27*b4 + 63*b3 - 54*b2 - 9*b1 - 6651) * q^69 + (-58*b7 - 165*b6 + 73*b5 + 65*b4 - 163*b3 - 362*b2 - 1199*b1 - 2527) * q^71 + (-81*b3 - 81*b2 - 648*b1 - 2025) * q^72 + (-8*b7 - 68*b6 + 122*b5 + 22*b4 - 242*b3 - 202*b2 - 720*b1 - 8060) * q^73 + (-17*b7 - 33*b6 - 70*b5 - 84*b4 - 267*b3 - 708*b2 - 5071*b1 - 21670) * q^74 + (-120*b7 - 418*b6 + 290*b5 - 22*b4 + 265*b3 - 643*b2 + 2562*b1 + 3023) * q^76 + (121*b6 + 121*b2 + 363*b1 - 1089) * q^77 + (-9*b7 + 99*b6 - 54*b5 - 72*b4 - 63*b3 - 684*b2 + 900*b1 - 16335) * q^78 + (126*b7 + 198*b6 - 111*b5 - 37*b4 + 175*b3 + 475*b2 - 3048*b1 - 2535) * q^79 + 6561 * q^81 + (73*b7 - 57*b6 - 20*b5 - 8*b4 + 188*b3 - 88*b2 + 5170*b1 + 6444) * q^82 + (-165*b7 - 184*b6 + 86*b5 + 49*b4 + 179*b3 - 339*b2 + 129*b1 - 12368) * q^83 + (-117*b7 - 99*b6 + 162*b5 + 36*b4 + 81*b3 + 216*b2 - 1233*b1 - 6552) * q^84 + (41*b7 + 73*b6 + 254*b5 - 124*b4 + 419*b3 - 354*b2 + 5232*b1 + 5459) * q^86 + (-9*b7 - 54*b6 - 198*b5 + 63*b4 + 45*b3 - 135*b2 + 909*b1 + 1134) * q^87 + (121*b3 + 121*b2 + 968*b1 + 3025) * q^88 + (131*b7 + 102*b6 + 16*b5 + 11*b4 - 235*b3 + 929*b2 - 7661*b1 - 2630) * q^89 + (36*b7 - 331*b6 + 97*b5 + b4 + 33*b3 - 648*b2 + 1647*b1 + 9128) * q^91 + (104*b7 - 62*b6 + 46*b5 + 22*b4 - 364*b3 - 745*b2 + 655*b1 - 11997) * q^92 + (81*b7 + 189*b6 + 81*b5 - 198*b3 - 333*b2 + 396*b1 - 5382) * q^93 + (-105*b7 - 369*b6 + 266*b5 + 180*b4 - 203*b3 + 876*b2 - 7269*b1 - 20700) * q^94 + (-36*b7 + 72*b6 - 180*b5 + 117*b3 - 99*b2 + 4266*b1 - 4761) * q^96 + (-208*b7 + 446*b6 + 298*b5 + 18*b4 - 30*b3 + 452*b2 + 4322*b1 - 18577) * q^97 + (-247*b7 - 437*b6 + 212*b5 + 12*b4 + 56*b3 - 747*b2 + 7567*b1 - 38573) * q^98 - 9801 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 9 q^{2} + 72 q^{3} + 107 q^{4} - 81 q^{6} + 66 q^{7} - 207 q^{8} + 648 q^{9}+O(q^{10})$$ 8 * q - 9 * q^2 + 72 * q^3 + 107 * q^4 - 81 * q^6 + 66 * q^7 - 207 * q^8 + 648 * q^9 $$8 q - 9 q^{2} + 72 q^{3} + 107 q^{4} - 81 q^{6} + 66 q^{7} - 207 q^{8} + 648 q^{9} - 968 q^{11} + 963 q^{12} - 382 q^{13} + 1048 q^{14} - 325 q^{16} - 288 q^{17} - 729 q^{18} - 988 q^{19} + 594 q^{21} + 1089 q^{22} - 5972 q^{23} - 1863 q^{24} - 14579 q^{26} + 5832 q^{27} - 5942 q^{28} + 1032 q^{29} - 4682 q^{31} - 3863 q^{32} - 8712 q^{33} + 2206 q^{34} + 8667 q^{36} + 17200 q^{37} - 11011 q^{38} - 3438 q^{39} - 13220 q^{41} + 9432 q^{42} - 22872 q^{43} - 12947 q^{44} + 9101 q^{46} - 6700 q^{47} - 2925 q^{48} - 43466 q^{49} - 2592 q^{51} - 5009 q^{52} - 6224 q^{53} - 6561 q^{54} + 20992 q^{56} - 8892 q^{57} - 33015 q^{58} - 77556 q^{59} + 11554 q^{61} - 12135 q^{62} + 5346 q^{63} - 149917 q^{64} + 9801 q^{66} - 20894 q^{67} - 91776 q^{68} - 53748 q^{69} - 21648 q^{71} - 16767 q^{72} - 64660 q^{73} - 179522 q^{74} + 24401 q^{76} - 7986 q^{77} - 131211 q^{78} - 22660 q^{79} + 52488 q^{81} + 56080 q^{82} - 100390 q^{83} - 53478 q^{84} + 47271 q^{86} + 9288 q^{87} + 25047 q^{88} - 25578 q^{89} + 73250 q^{91} - 95311 q^{92} - 42138 q^{93} - 170120 q^{94} - 34767 q^{96} - 142828 q^{97} - 303397 q^{98} - 78408 q^{99}+O(q^{100})$$ 8 * q - 9 * q^2 + 72 * q^3 + 107 * q^4 - 81 * q^6 + 66 * q^7 - 207 * q^8 + 648 * q^9 - 968 * q^11 + 963 * q^12 - 382 * q^13 + 1048 * q^14 - 325 * q^16 - 288 * q^17 - 729 * q^18 - 988 * q^19 + 594 * q^21 + 1089 * q^22 - 5972 * q^23 - 1863 * q^24 - 14579 * q^26 + 5832 * q^27 - 5942 * q^28 + 1032 * q^29 - 4682 * q^31 - 3863 * q^32 - 8712 * q^33 + 2206 * q^34 + 8667 * q^36 + 17200 * q^37 - 11011 * q^38 - 3438 * q^39 - 13220 * q^41 + 9432 * q^42 - 22872 * q^43 - 12947 * q^44 + 9101 * q^46 - 6700 * q^47 - 2925 * q^48 - 43466 * q^49 - 2592 * q^51 - 5009 * q^52 - 6224 * q^53 - 6561 * q^54 + 20992 * q^56 - 8892 * q^57 - 33015 * q^58 - 77556 * q^59 + 11554 * q^61 - 12135 * q^62 + 5346 * q^63 - 149917 * q^64 + 9801 * q^66 - 20894 * q^67 - 91776 * q^68 - 53748 * q^69 - 21648 * q^71 - 16767 * q^72 - 64660 * q^73 - 179522 * q^74 + 24401 * q^76 - 7986 * q^77 - 131211 * q^78 - 22660 * q^79 + 52488 * q^81 + 56080 * q^82 - 100390 * q^83 - 53478 * q^84 + 47271 * q^86 + 9288 * q^87 + 25047 * q^88 - 25578 * q^89 + 73250 * q^91 - 95311 * q^92 - 42138 * q^93 - 170120 * q^94 - 34767 * q^96 - 142828 * q^97 - 303397 * q^98 - 78408 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 44$$ v^2 + v - 44 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2\nu^{2} - 70\nu - 44$$ v^3 + 2*v^2 - 70*v - 44 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} - 14\nu^{5} + 164\nu^{4} + 1600\nu^{3} - 7927\nu^{2} - 36334\nu + 55536 ) / 160$$ (-v^6 - 14*v^5 + 164*v^4 + 1600*v^3 - 7927*v^2 - 36334*v + 55536) / 160 $$\beta_{5}$$ $$=$$ $$( \nu^{7} - 160\nu^{5} + 176\nu^{4} + 7407\nu^{3} - 13084\nu^{2} - 85572\nu + 99104 ) / 320$$ (v^7 - 160*v^5 + 176*v^4 + 7407*v^3 - 13084*v^2 - 85572*v + 99104) / 320 $$\beta_{6}$$ $$=$$ $$( \nu^{7} - 2\nu^{6} - 188\nu^{5} + 344\nu^{4} + 10287\nu^{3} - 15018\nu^{2} - 146720\nu + 89856 ) / 320$$ (v^7 - 2*v^6 - 188*v^5 + 344*v^4 + 10287*v^3 - 15018*v^2 - 146720*v + 89856) / 320 $$\beta_{7}$$ $$=$$ $$( -3\nu^{7} - 4\nu^{6} + 504\nu^{5} + 208\nu^{4} - 24861\nu^{3} + 6664\nu^{2} + 314100\nu - 129568 ) / 320$$ (-3*v^7 - 4*v^6 + 504*v^5 + 208*v^4 - 24861*v^3 + 6664*v^2 + 314100*v - 129568) / 320
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta _1 + 44$$ b2 - b1 + 44 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2\beta_{2} + 72\beta _1 - 44$$ b3 - 2*b2 + 72*b1 - 44 $$\nu^{4}$$ $$=$$ $$-2\beta_{6} + 2\beta_{5} + 2\beta_{4} - 2\beta_{3} + 91\beta_{2} - 159\beta _1 + 3164$$ -2*b6 + 2*b5 + 2*b4 - 2*b3 + 91*b2 - 159*b1 + 3164 $$\nu^{5}$$ $$=$$ $$4\beta_{7} + 2\beta_{6} + 10\beta_{5} - 10\beta_{4} + 115\beta_{3} - 306\beta_{2} + 5750\beta _1 - 6972$$ 4*b7 + 2*b6 + 10*b5 - 10*b4 + 115*b3 - 306*b2 + 5750*b1 - 6972 $$\nu^{6}$$ $$=$$ $$-56\beta_{7} - 356\beta_{6} + 188\beta_{5} + 308\beta_{4} - 338\beta_{3} + 8081\beta_{2} - 19783\beta _1 + 252852$$ -56*b7 - 356*b6 + 188*b5 + 308*b4 - 338*b3 + 8081*b2 - 19783*b1 + 252852 $$\nu^{7}$$ $$=$$ $$640 \beta_{7} + 672 \beta_{6} + 1568 \beta_{5} - 1952 \beta_{4} + 11345 \beta_{3} - 37078 \beta_{2} + 487168 \beta _1 - 869884$$ 640*b7 + 672*b6 + 1568*b5 - 1952*b4 + 11345*b3 - 37078*b2 + 487168*b1 - 869884

## 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
 8.70245 7.96197 5.43413 0.894159 0.292443 −4.15857 −8.09383 −10.0327
−9.70245 9.00000 62.1375 0 −87.3220 79.4231 −292.408 81.0000 0
1.2 −8.96197 9.00000 48.3168 0 −80.6577 −191.454 −146.231 81.0000 0
1.3 −6.43413 9.00000 9.39797 0 −57.9071 9.96708 145.424 81.0000 0
1.4 −1.89416 9.00000 −28.4122 0 −17.0474 191.987 114.430 81.0000 0
1.5 −1.29244 9.00000 −30.3296 0 −11.6320 −91.7671 80.5574 81.0000 0
1.6 3.15857 9.00000 −22.0234 0 28.4272 35.7175 −170.637 81.0000 0
1.7 7.09383 9.00000 18.3224 0 63.8444 −4.62438 −97.0268 81.0000 0
1.8 9.03274 9.00000 49.5905 0 81.2947 36.7509 158.890 81.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$1$$

## 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 825.6.a.p 8
5.b even 2 1 825.6.a.q yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.6.a.p 8 1.a even 1 1 trivial
825.6.a.q yes 8 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 9T_{2}^{7} - 141T_{2}^{6} - 1251T_{2}^{5} + 5160T_{2}^{4} + 45922T_{2}^{3} - 22268T_{2}^{2} - 305880T_{2} - 277200$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(825))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 9 T^{7} - 141 T^{6} + \cdots - 277200$$
$3$ $$(T - 9)^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 66 T^{7} + \cdots - 16208419192764$$
$11$ $$(T + 121)^{8}$$
$13$ $$T^{8} + 382 T^{7} + \cdots + 29\!\cdots\!92$$
$17$ $$T^{8} + 288 T^{7} + \cdots - 29\!\cdots\!00$$
$19$ $$T^{8} + 988 T^{7} + \cdots - 27\!\cdots\!23$$
$23$ $$T^{8} + 5972 T^{7} + \cdots - 31\!\cdots\!28$$
$29$ $$T^{8} - 1032 T^{7} + \cdots + 72\!\cdots\!84$$
$31$ $$T^{8} + 4682 T^{7} + \cdots - 95\!\cdots\!00$$
$37$ $$T^{8} - 17200 T^{7} + \cdots - 29\!\cdots\!64$$
$41$ $$T^{8} + 13220 T^{7} + \cdots - 33\!\cdots\!96$$
$43$ $$T^{8} + 22872 T^{7} + \cdots + 68\!\cdots\!72$$
$47$ $$T^{8} + 6700 T^{7} + \cdots + 21\!\cdots\!00$$
$53$ $$T^{8} + 6224 T^{7} + \cdots - 55\!\cdots\!44$$
$59$ $$T^{8} + 77556 T^{7} + \cdots + 27\!\cdots\!00$$
$61$ $$T^{8} - 11554 T^{7} + \cdots - 59\!\cdots\!00$$
$67$ $$T^{8} + 20894 T^{7} + \cdots + 45\!\cdots\!00$$
$71$ $$T^{8} + 21648 T^{7} + \cdots + 39\!\cdots\!88$$
$73$ $$T^{8} + 64660 T^{7} + \cdots + 59\!\cdots\!56$$
$79$ $$T^{8} + 22660 T^{7} + \cdots + 21\!\cdots\!12$$
$83$ $$T^{8} + 100390 T^{7} + \cdots + 27\!\cdots\!00$$
$89$ $$T^{8} + 25578 T^{7} + \cdots + 21\!\cdots\!32$$
$97$ $$T^{8} + 142828 T^{7} + \cdots - 13\!\cdots\!75$$