# Properties

 Label 225.4.e.c Level $225$ Weight $4$ Character orbit 225.e Analytic conductor $13.275$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.15759792.1 Defining polynomial: $$x^{6} - 3x^{5} + 16x^{4} - 27x^{3} + 52x^{2} - 39x + 9$$ x^6 - 3*x^5 + 16*x^4 - 27*x^3 + 52*x^2 - 39*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 45) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{5} - \beta_{2} - 1) q^{3} + (2 \beta_{5} - 3 \beta_{4} + 4 \beta_{3}) q^{4} + ( - \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 17) q^{6} + ( - \beta_{5} + 6 \beta_{4} - 16 \beta_{3} + 6 \beta_{2} + \beta_1 - 16) q^{7} + ( - 3 \beta_{2} - 3 \beta_1 + 9) q^{8} + ( - 2 \beta_{5} + 5 \beta_{4} - 22 \beta_{3} + 7 \beta_{2} + 4 \beta_1 - 2) q^{9}+O(q^{10})$$ q + (-b5 + b1) * q^2 + (b5 - b2 - 1) * q^3 + (2*b5 - 3*b4 + 4*b3) * q^4 + (-b4 + 5*b3 - 4*b2 - 2*b1 + 17) * q^6 + (-b5 + 6*b4 - 16*b3 + 6*b2 + b1 - 16) * q^7 + (-3*b2 - 3*b1 + 9) * q^8 + (-2*b5 + 5*b4 - 22*b3 + 7*b2 + 4*b1 - 2) * q^9 $$q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{5} - \beta_{2} - 1) q^{3} + (2 \beta_{5} - 3 \beta_{4} + 4 \beta_{3}) q^{4} + ( - \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 17) q^{6} + ( - \beta_{5} + 6 \beta_{4} - 16 \beta_{3} + 6 \beta_{2} + \beta_1 - 16) q^{7} + ( - 3 \beta_{2} - 3 \beta_1 + 9) q^{8} + ( - 2 \beta_{5} + 5 \beta_{4} - 22 \beta_{3} + 7 \beta_{2} + 4 \beta_1 - 2) q^{9} + ( - 14 \beta_{5} + 9 \beta_{4} - 3 \beta_{3} + 9 \beta_{2} + 14 \beta_1 - 3) q^{11} + ( - 17 \beta_{5} + 10 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + 11 \beta_1 - 9) q^{12} + (14 \beta_{5} - 3 \beta_{4} - 17 \beta_{3}) q^{13} + (24 \beta_{5} + 3 \beta_{4} - 18 \beta_{3}) q^{14} + ( - 2 \beta_{5} - 18 \beta_{4} + 11 \beta_{3} - 18 \beta_{2} + 2 \beta_1 + 11) q^{16} + ( - 3 \beta_{2} + 2 \beta_1 + 57) q^{17} + (17 \beta_{5} - 5 \beta_{4} + 13 \beta_{3} - 4 \beta_{2} + 14 \beta_1 + 14) q^{18} + ( - 9 \beta_{2} - 10 \beta_1 - 55) q^{19} + (12 \beta_{5} - 3 \beta_{4} - 33 \beta_{3} - 36 \beta_1 - 51) q^{21} + (40 \beta_{5} - 33 \beta_{4} + 123 \beta_{3}) q^{22} + (9 \beta_{5} - 36 \beta_{4} - 48 \beta_{3}) q^{23} + (18 \beta_{5} - 6 \beta_{4} + 21 \beta_{3} + 3 \beta_{2} + 6 \beta_1 + 3) q^{24} + ( - 39 \beta_{2} - 14 \beta_1 + 153) q^{26} + (24 \beta_{5} + 9 \beta_{4} - 27 \beta_{3} - 24 \beta_{2} - 45 \beta_1 - 42) q^{27} + ( - 27 \beta_{2} - 19 \beta_1 + 175) q^{28} + (14 \beta_{5} - 30 \beta_{4} + 117 \beta_{3} - 30 \beta_{2} - 14 \beta_1 + 117) q^{29} + (62 \beta_{5} - 33 \beta_{4} - 127 \beta_{3}) q^{31} + ( - 49 \beta_{5} + 42 \beta_{3}) q^{32} + (18 \beta_{5} - 38 \beta_{4} - 8 \beta_{3} - 71 \beta_{2} - 58 \beta_1 + 115) q^{33} + ( - 56 \beta_{5} - 9 \beta_{4} + 39 \beta_{3} - 9 \beta_{2} + 56 \beta_1 + 39) q^{34} + (26 \beta_{5} - 62 \beta_{4} + 28 \beta_{3} - 52 \beta_{2} - 46 \beta_1 + 191) q^{36} + (51 \beta_{2} + 24 \beta_1 - 143) q^{37} + (26 \beta_{5} + 21 \beta_{4} - 75 \beta_{3} + 21 \beta_{2} - 26 \beta_1 - 75) q^{38} + ( - 20 \beta_{5} + 79 \beta_{4} - 179 \beta_{3} + 39 \beta_{2} + \cdots - 153) q^{39}+ \cdots + (160 \beta_{5} + 53 \beta_{4} + 575 \beta_{3} + 172 \beta_{2} + 208 \beta_1 + 208) q^{99}+O(q^{100})$$ q + (-b5 + b1) * q^2 + (b5 - b2 - 1) * q^3 + (2*b5 - 3*b4 + 4*b3) * q^4 + (-b4 + 5*b3 - 4*b2 - 2*b1 + 17) * q^6 + (-b5 + 6*b4 - 16*b3 + 6*b2 + b1 - 16) * q^7 + (-3*b2 - 3*b1 + 9) * q^8 + (-2*b5 + 5*b4 - 22*b3 + 7*b2 + 4*b1 - 2) * q^9 + (-14*b5 + 9*b4 - 3*b3 + 9*b2 + 14*b1 - 3) * q^11 + (-17*b5 + 10*b4 + 4*b3 + 3*b2 + 11*b1 - 9) * q^12 + (14*b5 - 3*b4 - 17*b3) * q^13 + (24*b5 + 3*b4 - 18*b3) * q^14 + (-2*b5 - 18*b4 + 11*b3 - 18*b2 + 2*b1 + 11) * q^16 + (-3*b2 + 2*b1 + 57) * q^17 + (17*b5 - 5*b4 + 13*b3 - 4*b2 + 14*b1 + 14) * q^18 + (-9*b2 - 10*b1 - 55) * q^19 + (12*b5 - 3*b4 - 33*b3 - 36*b1 - 51) * q^21 + (40*b5 - 33*b4 + 123*b3) * q^22 + (9*b5 - 36*b4 - 48*b3) * q^23 + (18*b5 - 6*b4 + 21*b3 + 3*b2 + 6*b1 + 3) * q^24 + (-39*b2 - 14*b1 + 153) * q^26 + (24*b5 + 9*b4 - 27*b3 - 24*b2 - 45*b1 - 42) * q^27 + (-27*b2 - 19*b1 + 175) * q^28 + (14*b5 - 30*b4 + 117*b3 - 30*b2 - 14*b1 + 117) * q^29 + (62*b5 - 33*b4 - 127*b3) * q^31 + (-49*b5 + 42*b3) * q^32 + (18*b5 - 38*b4 - 8*b3 - 71*b2 - 58*b1 + 115) * q^33 + (-56*b5 - 9*b4 + 39*b3 - 9*b2 + 56*b1 + 39) * q^34 + (26*b5 - 62*b4 + 28*b3 - 52*b2 - 46*b1 + 191) * q^36 + (51*b2 + 24*b1 - 143) * q^37 + (26*b5 + 21*b4 - 75*b3 + 21*b2 - 26*b1 - 75) * q^38 + (-20*b5 + 79*b4 - 179*b3 + 39*b2 + 14*b1 - 153) * q^39 + (40*b5 - 69*b4 + 72*b3) * q^41 + (-21*b5 + 108*b4 - 432*b3 + 75*b2 + 27*b1 - 303) * q^42 + (-8*b5 - 87*b4 - 169*b3 - 87*b2 + 8*b1 - 169) * q^43 + (-15*b2 - 124*b1 + 291) * q^44 + (9*b2 - 6*b1 - 72) * q^46 + (59*b5 - 15*b4 + 207*b3 - 15*b2 - 59*b1 + 207) * q^47 + (-36*b5 + 41*b4 + 161*b3 + 17*b2 + 61*b1 + 275) * q^48 + (-26*b5 - 111*b4 + 189*b3) * q^49 + (56*b5 + 9*b4 - 39*b3 - 50*b2 + 6*b1 - 20) * q^51 + (-108*b5 - 21*b4 - 109*b3 - 21*b2 + 108*b1 - 109) * q^52 + (-24*b2 - 70*b1 + 228) * q^53 + (-72*b5 + 111*b4 - 420*b3 + 30*b2 + 60*b1 - 87) * q^54 + (-48*b5 + 54*b4 - 237*b3 + 54*b2 + 48*b1 - 237) * q^56 + (-26*b5 - 21*b4 + 75*b3 + 92*b2 + 18*b1 + 86) * q^57 + (-175*b5 + 12*b4 - 18*b3) * q^58 + (-82*b5 - 108*b4 - 36*b3) * q^59 + (20*b5 + 75*b4 + 448*b3 + 75*b2 - 20*b1 + 448) * q^61 + (-153*b2 - 30*b1 + 579) * q^62 + (21*b5 - 21*b4 + 303*b3 + 120*b2 - 6*b1 - 258) * q^63 + (3*b2 + 72*b1 - 500) * q^64 + (-302*b5 + 103*b4 - 341*b3 + 87*b2 + 236*b1 - 315) * q^66 + (-41*b5 - 21*b4 + 637*b3) * q^67 + (80*b5 - 153*b4 + 261*b3) * q^68 + (-78*b5 + 156*b4 + 399*b3 - 9*b2 + 6*b1 + 72) * q^69 + (-249*b2 - 76*b1 + 81) * q^71 + (-87*b5 + 78*b4 - 300*b3 + 30*b2 + 57*b1 - 186) * q^72 + (-192*b2 - 44*b1 + 172) * q^73 + (242*b5 - 21*b4 + 33*b3 - 21*b2 - 242*b1 + 33) * q^74 + (-36*b5 + 171*b4 + 23*b3) * q^76 + (240*b5 - 3*b4 + 237*b3) * q^77 + (220*b5 - 3*b4 - 27*b3 - 22*b2 + 78*b1 + 128) * q^78 + (112*b5 - 192*b4 + 154*b3 - 192*b2 - 112*b1 + 154) * q^79 + (78*b5 - 6*b4 - 87*b3 + 240*b2 + 60*b1 - 300) * q^81 + (-51*b2 - 221*b1 + 135) * q^82 + (-105*b5 + 354*b4 - 360*b3 + 354*b2 + 105*b1 - 360) * q^83 + (240*b5 - 30*b4 + 93*b3 - 75*b2 + 54*b1 - 27) * q^84 + (98*b5 - 111*b4 + 531*b3) * q^86 + (-60*b5 - 13*b4 + 83*b3 - b2 + 235*b1 + 65) * q^87 + (-234*b5 + 93*b4 - 429*b3 + 93*b2 + 234*b1 - 429) * q^88 + (240*b2 + 168*b1 + 549) * q^89 + (135*b2 - 286*b1 - 377) * q^91 + (141*b5 - 261*b4 - 501*b3 - 261*b2 - 141*b1 - 501) * q^92 + (-96*b5 + 441*b4 - 465*b3 + 153*b2 + 30*b1 - 579) * q^93 + (-340*b5 + 162*b4 - 633*b3) * q^94 + (56*b5 - 238*b4 + 791*b3 - 147*b2 - 56*b1 + 588) * q^96 + (78*b5 - 60*b4 - 100*b3 - 60*b2 - 78*b1 - 100) * q^97 + (189*b2 - 248*b1 - 867) * q^98 + (160*b5 + 53*b4 + 575*b3 + 172*b2 + 208*b1 + 208) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{2} - 9 q^{3} - 11 q^{4} + 84 q^{6} - 43 q^{7} + 54 q^{8} + 57 q^{9}+O(q^{10})$$ 6 * q - q^2 - 9 * q^3 - 11 * q^4 + 84 * q^6 - 43 * q^7 + 54 * q^8 + 57 * q^9 $$6 q - q^{2} - 9 q^{3} - 11 q^{4} + 84 q^{6} - 43 q^{7} + 54 q^{8} + 57 q^{9} - 14 q^{11} - 75 q^{12} + 40 q^{13} + 27 q^{14} + 13 q^{16} + 332 q^{17} - 3 q^{18} - 328 q^{19} - 144 q^{21} - 376 q^{22} + 171 q^{23} - 63 q^{24} + 868 q^{26} - 162 q^{27} + 1034 q^{28} + 335 q^{29} + 352 q^{31} - 77 q^{32} + 708 q^{33} + 52 q^{34} + 1086 q^{36} - 804 q^{37} - 178 q^{38} - 390 q^{39} - 187 q^{41} - 513 q^{42} - 602 q^{43} + 1964 q^{44} - 402 q^{46} + 665 q^{47} + 1074 q^{48} - 430 q^{49} - 180 q^{51} - 456 q^{52} + 1460 q^{53} + 639 q^{54} - 705 q^{56} + 486 q^{57} + 217 q^{58} + 298 q^{59} + 1439 q^{61} + 3228 q^{62} - 2205 q^{63} - 3138 q^{64} - 966 q^{66} - 1849 q^{67} - 710 q^{68} - 873 q^{69} + 140 q^{71} - 261 q^{72} + 736 q^{73} + 320 q^{74} - 204 q^{76} - 948 q^{77} + 432 q^{78} + 382 q^{79} - 1251 q^{81} + 1150 q^{82} - 831 q^{83} - 909 q^{84} - 1580 q^{86} - 258 q^{87} - 1428 q^{88} + 3438 q^{89} - 1420 q^{91} - 1623 q^{92} - 2178 q^{93} + 2077 q^{94} + 1155 q^{96} - 282 q^{97} - 4328 q^{98} - 762 q^{99}+O(q^{100})$$ 6 * q - q^2 - 9 * q^3 - 11 * q^4 + 84 * q^6 - 43 * q^7 + 54 * q^8 + 57 * q^9 - 14 * q^11 - 75 * q^12 + 40 * q^13 + 27 * q^14 + 13 * q^16 + 332 * q^17 - 3 * q^18 - 328 * q^19 - 144 * q^21 - 376 * q^22 + 171 * q^23 - 63 * q^24 + 868 * q^26 - 162 * q^27 + 1034 * q^28 + 335 * q^29 + 352 * q^31 - 77 * q^32 + 708 * q^33 + 52 * q^34 + 1086 * q^36 - 804 * q^37 - 178 * q^38 - 390 * q^39 - 187 * q^41 - 513 * q^42 - 602 * q^43 + 1964 * q^44 - 402 * q^46 + 665 * q^47 + 1074 * q^48 - 430 * q^49 - 180 * q^51 - 456 * q^52 + 1460 * q^53 + 639 * q^54 - 705 * q^56 + 486 * q^57 + 217 * q^58 + 298 * q^59 + 1439 * q^61 + 3228 * q^62 - 2205 * q^63 - 3138 * q^64 - 966 * q^66 - 1849 * q^67 - 710 * q^68 - 873 * q^69 + 140 * q^71 - 261 * q^72 + 736 * q^73 + 320 * q^74 - 204 * q^76 - 948 * q^77 + 432 * q^78 + 382 * q^79 - 1251 * q^81 + 1150 * q^82 - 831 * q^83 - 909 * q^84 - 1580 * q^86 - 258 * q^87 - 1428 * q^88 + 3438 * q^89 - 1420 * q^91 - 1623 * q^92 - 2178 * q^93 + 2077 * q^94 + 1155 * q^96 - 282 * q^97 - 4328 * q^98 - 762 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 16x^{4} - 27x^{3} + 52x^{2} - 39x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 4$$ v^2 - v + 4 $$\beta_{2}$$ $$=$$ $$( -\nu^{4} + 2\nu^{3} - 11\nu^{2} + 10\nu - 12 ) / 3$$ (-v^4 + 2*v^3 - 11*v^2 + 10*v - 12) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 30\nu^{3} - 40\nu^{2} + 88\nu - 39 ) / 3$$ (2*v^5 - 5*v^4 + 30*v^3 - 40*v^2 + 88*v - 39) / 3 $$\beta_{4}$$ $$=$$ $$( -7\nu^{5} + 18\nu^{4} - 103\nu^{3} + 141\nu^{2} - 289\nu + 126 ) / 3$$ (-7*v^5 + 18*v^4 - 103*v^3 + 141*v^2 - 289*v + 126) / 3 $$\beta_{5}$$ $$=$$ $$( -8\nu^{5} + 20\nu^{4} - 117\nu^{3} + 157\nu^{2} - 334\nu + 147 ) / 3$$ (-8*v^5 + 20*v^4 - 117*v^3 + 157*v^2 - 334*v + 147) / 3
 $$\nu$$ $$=$$ $$( -2\beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 3$$ (-2*b5 + 2*b4 - b3 + b2 + b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( -2\beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + 4\beta _1 - 11 ) / 3$$ (-2*b5 + 2*b4 - b3 + b2 + 4*b1 - 11) / 3 $$\nu^{3}$$ $$=$$ $$( 13\beta_{5} - 10\beta_{4} + 17\beta_{3} - 5\beta_{2} - 2\beta _1 - 8 ) / 3$$ (13*b5 - 10*b4 + 17*b3 - 5*b2 - 2*b1 - 8) / 3 $$\nu^{4}$$ $$=$$ $$( 28\beta_{5} - 22\beta_{4} + 35\beta_{3} - 20\beta_{2} - 38\beta _1 + 79 ) / 3$$ (28*b5 - 22*b4 + 35*b3 - 20*b2 - 38*b1 + 79) / 3 $$\nu^{5}$$ $$=$$ $$( -77\beta_{5} + 47\beta_{4} - 139\beta_{3} + \beta_{2} - 29\beta _1 + 112 ) / 3$$ (-77*b5 + 47*b4 - 139*b3 + b2 - 29*b1 + 112) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 0.5 − 2.88506i 0.5 + 1.98116i 0.5 + 0.0378788i 0.5 + 2.88506i 0.5 − 1.98116i 0.5 − 0.0378788i
−2.28679 3.96084i −3.36330 + 3.96084i −6.45882 + 11.1870i 0 23.3794 + 4.26387i −10.0573 17.4197i 22.4912 −4.37646 26.6429i 0
76.2 −0.0874923 0.151541i −5.19394 + 0.151541i 3.98469 6.90169i 0 0.477395 + 0.773837i 4.23186 + 7.32979i −2.79440 26.9541 1.57419i 0
76.3 1.87428 + 3.24635i 4.05724 3.24635i −3.02587 + 5.24096i 0 18.1432 + 7.08665i −15.6746 27.1492i 7.30318 5.92239 26.3425i 0
151.1 −2.28679 + 3.96084i −3.36330 3.96084i −6.45882 11.1870i 0 23.3794 4.26387i −10.0573 + 17.4197i 22.4912 −4.37646 + 26.6429i 0
151.2 −0.0874923 + 0.151541i −5.19394 0.151541i 3.98469 + 6.90169i 0 0.477395 0.773837i 4.23186 7.32979i −2.79440 26.9541 + 1.57419i 0
151.3 1.87428 3.24635i 4.05724 + 3.24635i −3.02587 5.24096i 0 18.1432 7.08665i −15.6746 + 27.1492i 7.30318 5.92239 + 26.3425i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 151.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.e.c 6
5.b even 2 1 45.4.e.b 6
5.c odd 4 2 225.4.k.c 12
9.c even 3 1 inner 225.4.e.c 6
9.c even 3 1 2025.4.a.s 3
9.d odd 6 1 2025.4.a.q 3
15.d odd 2 1 135.4.e.b 6
45.h odd 6 1 135.4.e.b 6
45.h odd 6 1 405.4.a.j 3
45.j even 6 1 45.4.e.b 6
45.j even 6 1 405.4.a.h 3
45.k odd 12 2 225.4.k.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.b 6 5.b even 2 1
45.4.e.b 6 45.j even 6 1
135.4.e.b 6 15.d odd 2 1
135.4.e.b 6 45.h odd 6 1
225.4.e.c 6 1.a even 1 1 trivial
225.4.e.c 6 9.c even 3 1 inner
225.4.k.c 12 5.c odd 4 2
225.4.k.c 12 45.k odd 12 2
405.4.a.h 3 45.j even 6 1
405.4.a.j 3 45.h odd 6 1
2025.4.a.q 3 9.d odd 6 1
2025.4.a.s 3 9.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + T_{2}^{5} + 18T_{2}^{4} - 11T_{2}^{3} + 292T_{2}^{2} + 51T_{2} + 9$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{5} + 18 T^{4} - 11 T^{3} + \cdots + 9$$
$3$ $$T^{6} + 9 T^{5} + 12 T^{4} + \cdots + 19683$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 43 T^{5} + 1654 T^{4} + \cdots + 28483569$$
$11$ $$T^{6} + 14 T^{5} + \cdots + 1896428304$$
$13$ $$T^{6} - 40 T^{5} + \cdots + 5679732496$$
$17$ $$(T^{3} - 166 T^{2} + 8920 T - 156324)^{2}$$
$19$ $$(T^{3} + 164 T^{2} + 7292 T + 57316)^{2}$$
$23$ $$T^{6} - 171 T^{5} + \cdots + 3746541681$$
$29$ $$T^{6} - 335 T^{5} + \cdots + 11463342489$$
$31$ $$T^{6} - 352 T^{5} + \cdots + 97238137683600$$
$37$ $$(T^{3} + 402 T^{2} + 24708 T - 3335284)^{2}$$
$41$ $$T^{6} + 187 T^{5} + \cdots + 52247959475625$$
$43$ $$T^{6} + \cdots + 238533321586576$$
$47$ $$T^{6} - 665 T^{5} + \cdots + 6187571675289$$
$53$ $$(T^{3} - 730 T^{2} + 106300 T - 3250536)^{2}$$
$59$ $$T^{6} - 298 T^{5} + \cdots + 16\!\cdots\!16$$
$61$ $$T^{6} - 1439 T^{5} + \cdots + 30\!\cdots\!09$$
$67$ $$T^{6} + 1849 T^{5} + \cdots + 43\!\cdots\!81$$
$71$ $$(T^{3} - 70 T^{2} - 685460 T + 223775052)^{2}$$
$73$ $$(T^{3} - 368 T^{2} - 372928 T + 134927744)^{2}$$
$79$ $$T^{6} - 382 T^{5} + \cdots + 19\!\cdots\!56$$
$83$ $$T^{6} + 831 T^{5} + \cdots + 91\!\cdots\!41$$
$89$ $$(T^{3} - 1719 T^{2} + 238491 T + 125506395)^{2}$$
$97$ $$T^{6} + \cdots + 285551326213696$$