# Properties

 Label 225.4.k.a Level $225$ Weight $4$ Character orbit 225.k Analytic conductor $13.275$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.303595776.1 Defining polynomial: $$x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81$$ x^8 + 5*x^6 + 16*x^4 + 45*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{7} q^{2} + (2 \beta_{7} + 2 \beta_{5} + \beta_{4} + 3 \beta_1) q^{3} + ( - \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{4} + (\beta_{6} + 7 \beta_{3} + \beta_{2} - 9) q^{6} + ( - \beta_{7} - 5 \beta_1) q^{7} + (\beta_{5} - 7 \beta_{4} + \beta_1) q^{8} + (3 \beta_{6} + 3 \beta_{3} - 6 \beta_{2} + 27) q^{9}+O(q^{10})$$ q - b7 * q^2 + (2*b7 + 2*b5 + b4 + 3*b1) * q^3 + (-b6 - b3 + b2 - 1) * q^4 + (b6 + 7*b3 + b2 - 9) * q^6 + (-b7 - 5*b1) * q^7 + (b5 - 7*b4 + b1) * q^8 + (3*b6 + 3*b3 - 6*b2 + 27) * q^9 $$q - \beta_{7} q^{2} + (2 \beta_{7} + 2 \beta_{5} + \beta_{4} + 3 \beta_1) q^{3} + ( - \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{4} + (\beta_{6} + 7 \beta_{3} + \beta_{2} - 9) q^{6} + ( - \beta_{7} - 5 \beta_1) q^{7} + (\beta_{5} - 7 \beta_{4} + \beta_1) q^{8} + (3 \beta_{6} + 3 \beta_{3} - 6 \beta_{2} + 27) q^{9} + (21 \beta_{3} - 5 \beta_{2} + 21) q^{11} + (2 \beta_{7} - 7 \beta_{5} + \beta_{4} - 15 \beta_1) q^{12} + (8 \beta_{7} - 52 \beta_{5} + 8 \beta_{4} + 8 \beta_1) q^{13} + (4 \beta_{6} - 4 \beta_{3} - 4 \beta_{2} + 4) q^{14} + (63 \beta_{3} - 7 \beta_{2} + 63) q^{16} + (51 \beta_{5} + 25 \beta_{4} + 51 \beta_1) q^{17} + ( - 24 \beta_{7} - 51 \beta_{5} - 3 \beta_{4} - 27 \beta_1) q^{18} + ( - 25 \beta_{6} - 30) q^{19} + ( - 9 \beta_{6} + 12 \beta_{3} + 6 \beta_{2} - 9) q^{21} + ( - 21 \beta_{7} - 61 \beta_{5} - 21 \beta_{4} - 21 \beta_1) q^{22} + (23 \beta_{7} - 122 \beta_{5} + 23 \beta_{4} + 23 \beta_1) q^{23} + ( - 6 \beta_{6} - 111 \beta_{3} - 9 \beta_{2} - 63) q^{24} + ( - 52 \beta_{6} - 64) q^{26} + (36 \beta_{7} + 117 \beta_{5} + 18 \beta_{4} + 135 \beta_1) q^{27} + (4 \beta_{5} - 4 \beta_{4} + 4 \beta_1) q^{28} + (182 \beta_{3} - 39 \beta_{2} + 182) q^{29} - 6 \beta_{3} q^{31} + ( - 7 \beta_{7} - 127 \beta_{5} - 7 \beta_{4} - 7 \beta_1) q^{32} + (11 \beta_{7} + 101 \beta_{5} + 37 \beta_{4} + 93 \beta_1) q^{33} + ( - 251 \beta_{3} + 51 \beta_{2} - 251) q^{34} + ( - 24 \beta_{6} + 3 \beta_{3} + 21 \beta_{2} + 27) q^{36} + ( - 324 \beta_{5} - 10 \beta_{4} - 324 \beta_1) q^{37} + (5 \beta_{7} + 200 \beta_1) q^{38} + (68 \beta_{6} - 40 \beta_{3} + 44 \beta_{2} + 144) q^{39} + (60 \beta_{6} + 149 \beta_{3} - 60 \beta_{2} + 60) q^{41} + (36 \beta_{5} - 12 \beta_{4} + 60 \beta_1) q^{42} + ( - 87 \beta_{7} - 92 \beta_1) q^{43} + ( - 21 \beta_{6} + 40) q^{44} + ( - 122 \beta_{6} - 184) q^{46} + ( - 11 \beta_{7} + 445 \beta_1) q^{47} + (49 \beta_{7} + 175 \beta_{5} + 119 \beta_{4} + 231 \beta_1) q^{48} + (9 \beta_{6} + 319 \beta_{3} - 9 \beta_{2} + 9) q^{49} + (76 \beta_{6} + 451 \beta_{3} - 77 \beta_{2} + 225) q^{51} + ( - 52 \beta_{7} - 64 \beta_1) q^{52} + ( - 62 \beta_{5} + 100 \beta_{4} - 62 \beta_1) q^{53} + (18 \beta_{6} + 45 \beta_{3} + 99 \beta_{2} - 243) q^{54} + (60 \beta_{3} + 36 \beta_{2} + 60) q^{56} + ( - 35 \beta_{7} + 190 \beta_{5} + 20 \beta_{4} - 240 \beta_1) q^{57} + ( - 182 \beta_{7} - 494 \beta_{5} - 182 \beta_{4} - 182 \beta_1) q^{58} + (49 \beta_{6} + 67 \beta_{3} - 49 \beta_{2} + 49) q^{59} + (26 \beta_{3} + 195 \beta_{2} + 26) q^{61} + (6 \beta_{5} + 6 \beta_{4} + 6 \beta_1) q^{62} + ( - 9 \beta_{7} - 36 \beta_{5} + 27 \beta_{4} - 117 \beta_1) q^{63} + ( - 71 \beta_{6} - 392) q^{64} + (19 \beta_{6} - 290 \beta_{3} + 82 \beta_{2} - 378) q^{66} + (184 \beta_{7} + 395 \beta_{5} + 184 \beta_{4} + 184 \beta_1) q^{67} + (51 \beta_{7} + 251 \beta_{5} + 51 \beta_{4} + 51 \beta_1) q^{68} + (168 \beta_{6} - 60 \beta_{3} + 99 \beta_{2} + 414) q^{69} + (110 \beta_{6} + 252) q^{71} + ( - 27 \beta_{7} + 189 \beta_{5} - 171 \beta_{4} - 171 \beta_1) q^{72} + (303 \beta_{5} - 205 \beta_{4} + 303 \beta_1) q^{73} + (404 \beta_{3} - 324 \beta_{2} + 404) q^{74} + (5 \beta_{6} - 195 \beta_{3} - 5 \beta_{2} + 5) q^{76} + (4 \beta_{7} + 44 \beta_{5} + 4 \beta_{4} + 4 \beta_1) q^{77} + ( - 76 \beta_{7} + 392 \beta_{5} + 40 \beta_{4} - 504 \beta_1) q^{78} + ( - 458 \beta_{3} + 76 \beta_{2} - 458) q^{79} + (135 \beta_{6} + 135 \beta_{3} - 270 \beta_{2} + 486) q^{81} + ( - 629 \beta_{5} - 149 \beta_{4} - 629 \beta_1) q^{82} + (453 \beta_{7} + 33 \beta_1) q^{83} + ( - 60 \beta_{3} - 12 \beta_{2} - 36) q^{84} + (5 \beta_{6} - 691 \beta_{3} - 5 \beta_{2} + 5) q^{86} + (104 \beta_{7} + 806 \beta_{5} + 325 \beta_{4} + 780 \beta_1) q^{87} + (107 \beta_{7} - 152 \beta_1) q^{88} + ( - 315 \beta_{6} + 375) q^{89} + ( - 92 \beta_{6} - 364) q^{91} + ( - 122 \beta_{7} - 184 \beta_1) q^{92} + (6 \beta_{7} + 6 \beta_{5} - 6 \beta_{4}) q^{93} + ( - 456 \beta_{6} - 544 \beta_{3} + 456 \beta_{2} - 456) q^{94} + (113 \beta_{6} - 310 \beta_{3} + 134 \beta_{2} - 126) q^{96} + ( - 131 \beta_{7} + 450 \beta_1) q^{97} + ( - 391 \beta_{5} - 319 \beta_{4} - 391 \beta_1) q^{98} + (111 \beta_{6} + 624 \beta_{3} - 168 \beta_{2} + 432) q^{99}+O(q^{100})$$ q - b7 * q^2 + (2*b7 + 2*b5 + b4 + 3*b1) * q^3 + (-b6 - b3 + b2 - 1) * q^4 + (b6 + 7*b3 + b2 - 9) * q^6 + (-b7 - 5*b1) * q^7 + (b5 - 7*b4 + b1) * q^8 + (3*b6 + 3*b3 - 6*b2 + 27) * q^9 + (21*b3 - 5*b2 + 21) * q^11 + (2*b7 - 7*b5 + b4 - 15*b1) * q^12 + (8*b7 - 52*b5 + 8*b4 + 8*b1) * q^13 + (4*b6 - 4*b3 - 4*b2 + 4) * q^14 + (63*b3 - 7*b2 + 63) * q^16 + (51*b5 + 25*b4 + 51*b1) * q^17 + (-24*b7 - 51*b5 - 3*b4 - 27*b1) * q^18 + (-25*b6 - 30) * q^19 + (-9*b6 + 12*b3 + 6*b2 - 9) * q^21 + (-21*b7 - 61*b5 - 21*b4 - 21*b1) * q^22 + (23*b7 - 122*b5 + 23*b4 + 23*b1) * q^23 + (-6*b6 - 111*b3 - 9*b2 - 63) * q^24 + (-52*b6 - 64) * q^26 + (36*b7 + 117*b5 + 18*b4 + 135*b1) * q^27 + (4*b5 - 4*b4 + 4*b1) * q^28 + (182*b3 - 39*b2 + 182) * q^29 - 6*b3 * q^31 + (-7*b7 - 127*b5 - 7*b4 - 7*b1) * q^32 + (11*b7 + 101*b5 + 37*b4 + 93*b1) * q^33 + (-251*b3 + 51*b2 - 251) * q^34 + (-24*b6 + 3*b3 + 21*b2 + 27) * q^36 + (-324*b5 - 10*b4 - 324*b1) * q^37 + (5*b7 + 200*b1) * q^38 + (68*b6 - 40*b3 + 44*b2 + 144) * q^39 + (60*b6 + 149*b3 - 60*b2 + 60) * q^41 + (36*b5 - 12*b4 + 60*b1) * q^42 + (-87*b7 - 92*b1) * q^43 + (-21*b6 + 40) * q^44 + (-122*b6 - 184) * q^46 + (-11*b7 + 445*b1) * q^47 + (49*b7 + 175*b5 + 119*b4 + 231*b1) * q^48 + (9*b6 + 319*b3 - 9*b2 + 9) * q^49 + (76*b6 + 451*b3 - 77*b2 + 225) * q^51 + (-52*b7 - 64*b1) * q^52 + (-62*b5 + 100*b4 - 62*b1) * q^53 + (18*b6 + 45*b3 + 99*b2 - 243) * q^54 + (60*b3 + 36*b2 + 60) * q^56 + (-35*b7 + 190*b5 + 20*b4 - 240*b1) * q^57 + (-182*b7 - 494*b5 - 182*b4 - 182*b1) * q^58 + (49*b6 + 67*b3 - 49*b2 + 49) * q^59 + (26*b3 + 195*b2 + 26) * q^61 + (6*b5 + 6*b4 + 6*b1) * q^62 + (-9*b7 - 36*b5 + 27*b4 - 117*b1) * q^63 + (-71*b6 - 392) * q^64 + (19*b6 - 290*b3 + 82*b2 - 378) * q^66 + (184*b7 + 395*b5 + 184*b4 + 184*b1) * q^67 + (51*b7 + 251*b5 + 51*b4 + 51*b1) * q^68 + (168*b6 - 60*b3 + 99*b2 + 414) * q^69 + (110*b6 + 252) * q^71 + (-27*b7 + 189*b5 - 171*b4 - 171*b1) * q^72 + (303*b5 - 205*b4 + 303*b1) * q^73 + (404*b3 - 324*b2 + 404) * q^74 + (5*b6 - 195*b3 - 5*b2 + 5) * q^76 + (4*b7 + 44*b5 + 4*b4 + 4*b1) * q^77 + (-76*b7 + 392*b5 + 40*b4 - 504*b1) * q^78 + (-458*b3 + 76*b2 - 458) * q^79 + (135*b6 + 135*b3 - 270*b2 + 486) * q^81 + (-629*b5 - 149*b4 - 629*b1) * q^82 + (453*b7 + 33*b1) * q^83 + (-60*b3 - 12*b2 - 36) * q^84 + (5*b6 - 691*b3 - 5*b2 + 5) * q^86 + (104*b7 + 806*b5 + 325*b4 + 780*b1) * q^87 + (107*b7 - 152*b1) * q^88 + (-315*b6 + 375) * q^89 + (-92*b6 - 364) * q^91 + (-122*b7 - 184*b1) * q^92 + (6*b7 + 6*b5 - 6*b4) * q^93 + (-456*b6 - 544*b3 + 456*b2 - 456) * q^94 + (113*b6 - 310*b3 + 134*b2 - 126) * q^96 + (-131*b7 + 450*b1) * q^97 + (-391*b5 - 319*b4 - 391*b1) * q^98 + (111*b6 + 624*b3 - 168*b2 + 432) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{4} - 102 q^{6} + 180 q^{9}+O(q^{10})$$ 8 * q + 2 * q^4 - 102 * q^6 + 180 * q^9 $$8 q + 2 q^{4} - 102 q^{6} + 180 q^{9} + 74 q^{11} + 24 q^{14} + 238 q^{16} - 140 q^{19} - 72 q^{21} - 54 q^{24} - 304 q^{26} + 650 q^{29} + 24 q^{31} - 902 q^{34} + 342 q^{36} + 1128 q^{39} - 476 q^{41} + 404 q^{44} - 984 q^{46} - 1258 q^{49} - 462 q^{51} - 1998 q^{54} + 312 q^{56} - 170 q^{59} + 494 q^{61} - 2852 q^{64} - 1776 q^{66} + 3078 q^{69} + 1576 q^{71} + 968 q^{74} + 790 q^{76} - 1680 q^{79} + 2268 q^{81} - 72 q^{84} + 2774 q^{86} + 4260 q^{89} - 2544 q^{91} + 1264 q^{94} + 48 q^{96} + 180 q^{99}+O(q^{100})$$ 8 * q + 2 * q^4 - 102 * q^6 + 180 * q^9 + 74 * q^11 + 24 * q^14 + 238 * q^16 - 140 * q^19 - 72 * q^21 - 54 * q^24 - 304 * q^26 + 650 * q^29 + 24 * q^31 - 902 * q^34 + 342 * q^36 + 1128 * q^39 - 476 * q^41 + 404 * q^44 - 984 * q^46 - 1258 * q^49 - 462 * q^51 - 1998 * q^54 + 312 * q^56 - 170 * q^59 + 494 * q^61 - 2852 * q^64 - 1776 * q^66 + 3078 * q^69 + 1576 * q^71 + 968 * q^74 + 790 * q^76 - 1680 * q^79 + 2268 * q^81 - 72 * q^84 + 2774 * q^86 + 4260 * q^89 - 2544 * q^91 + 1264 * q^94 + 48 * q^96 + 180 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432$$ (v^7 + 32*v^5 + 16*v^3 + 45*v) / 432 $$\beta_{2}$$ $$=$$ $$( \nu^{6} - 16\nu^{4} - 32\nu^{2} - 51 ) / 48$$ (v^6 - 16*v^4 - 32*v^2 - 51) / 48 $$\beta_{3}$$ $$=$$ $$( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144$$ (-5*v^6 - 16*v^4 - 80*v^2 - 225) / 144 $$\beta_{4}$$ $$=$$ $$( \nu^{7} + 8\nu^{5} + 40\nu^{3} + 165\nu ) / 72$$ (v^7 + 8*v^5 + 40*v^3 + 165*v) / 72 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 13\nu ) / 48$$ (v^7 + 13*v) / 48 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} - 5\nu^{4} - 7\nu^{2} - 27 ) / 9$$ (-v^6 - 5*v^4 - 7*v^2 - 27) / 9 $$\beta_{7}$$ $$=$$ $$( \nu^{7} + 2\nu^{5} + 10\nu^{3} - 3\nu ) / 18$$ (v^7 + 2*v^5 + 10*v^3 - 3*v) / 18
 $$\nu$$ $$=$$ $$( -\beta_{7} + 2\beta_{5} + \beta_{4} ) / 3$$ (-b7 + 2*b5 + b4) / 3 $$\nu^{2}$$ $$=$$ $$( 2\beta_{6} - 7\beta_{3} - \beta_{2} - 6 ) / 3$$ (2*b6 - 7*b3 - b2 - 6) / 3 $$\nu^{3}$$ $$=$$ $$( 4\beta_{7} - 11\beta_{5} + 2\beta_{4} - 9\beta_1 ) / 3$$ (4*b7 - 11*b5 + 2*b4 - 9*b1) / 3 $$\nu^{4}$$ $$=$$ $$( -5\beta_{6} + 13\beta_{3} - 5\beta_{2} ) / 3$$ (-5*b6 + 13*b3 - 5*b2) / 3 $$\nu^{5}$$ $$=$$ $$( -\beta_{7} - \beta_{5} - 2\beta_{4} + 45\beta_1 ) / 3$$ (-b7 - b5 - 2*b4 + 45*b1) / 3 $$\nu^{6}$$ $$=$$ $$( -16\beta_{6} - 16\beta_{3} + 32\beta_{2} - 39 ) / 3$$ (-16*b6 - 16*b3 + 32*b2 - 39) / 3 $$\nu^{7}$$ $$=$$ $$( 13\beta_{7} + 118\beta_{5} - 13\beta_{4} ) / 3$$ (13*b7 + 118*b5 - 13*b4) / 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.396143 + 1.68614i −1.26217 + 1.18614i 1.26217 − 1.18614i 0.396143 − 1.68614i −0.396143 − 1.68614i −1.26217 − 1.18614i 1.26217 + 1.18614i 0.396143 + 1.68614i
−2.92048 + 1.68614i 4.97494 + 1.50000i 1.68614 2.92048i 0 −17.0584 + 4.00772i 1.40965 0.813859i 15.6060i 22.5000 + 14.9248i 0
49.2 −2.05446 + 1.18614i 4.97494 1.50000i −1.18614 + 2.05446i 0 −8.44158 + 8.98266i −6.38458 + 3.68614i 24.6060i 22.5000 14.9248i 0
49.3 2.05446 1.18614i −4.97494 + 1.50000i −1.18614 + 2.05446i 0 −8.44158 + 8.98266i 6.38458 3.68614i 24.6060i 22.5000 14.9248i 0
49.4 2.92048 1.68614i −4.97494 1.50000i 1.68614 2.92048i 0 −17.0584 + 4.00772i −1.40965 + 0.813859i 15.6060i 22.5000 + 14.9248i 0
124.1 −2.92048 1.68614i 4.97494 1.50000i 1.68614 + 2.92048i 0 −17.0584 4.00772i 1.40965 + 0.813859i 15.6060i 22.5000 14.9248i 0
124.2 −2.05446 1.18614i 4.97494 + 1.50000i −1.18614 2.05446i 0 −8.44158 8.98266i −6.38458 3.68614i 24.6060i 22.5000 + 14.9248i 0
124.3 2.05446 + 1.18614i −4.97494 1.50000i −1.18614 2.05446i 0 −8.44158 8.98266i 6.38458 + 3.68614i 24.6060i 22.5000 + 14.9248i 0
124.4 2.92048 + 1.68614i −4.97494 + 1.50000i 1.68614 + 2.92048i 0 −17.0584 4.00772i −1.40965 0.813859i 15.6060i 22.5000 14.9248i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 124.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.k.a 8
5.b even 2 1 inner 225.4.k.a 8
5.c odd 4 1 45.4.e.a 4
5.c odd 4 1 225.4.e.a 4
9.c even 3 1 inner 225.4.k.a 8
15.e even 4 1 135.4.e.a 4
45.j even 6 1 inner 225.4.k.a 8
45.k odd 12 1 45.4.e.a 4
45.k odd 12 1 225.4.e.a 4
45.k odd 12 1 405.4.a.d 2
45.k odd 12 1 2025.4.a.l 2
45.l even 12 1 135.4.e.a 4
45.l even 12 1 405.4.a.e 2
45.l even 12 1 2025.4.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.a 4 5.c odd 4 1
45.4.e.a 4 45.k odd 12 1
135.4.e.a 4 15.e even 4 1
135.4.e.a 4 45.l even 12 1
225.4.e.a 4 5.c odd 4 1
225.4.e.a 4 45.k odd 12 1
225.4.k.a 8 1.a even 1 1 trivial
225.4.k.a 8 5.b even 2 1 inner
225.4.k.a 8 9.c even 3 1 inner
225.4.k.a 8 45.j even 6 1 inner
405.4.a.d 2 45.k odd 12 1
405.4.a.e 2 45.l even 12 1
2025.4.a.j 2 45.l even 12 1
2025.4.a.l 2 45.k odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 17T_{2}^{6} + 225T_{2}^{4} - 1088T_{2}^{2} + 4096$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 17 T^{6} + 225 T^{4} + \cdots + 4096$$
$3$ $$(T^{4} - 45 T^{2} + 729)^{2}$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 57 T^{6} + 3105 T^{4} + \cdots + 20736$$
$11$ $$(T^{4} - 37 T^{3} + 1233 T^{2} + \cdots + 18496)^{2}$$
$13$ $$T^{8} - 7328 T^{6} + \cdots + 46262633168896$$
$17$ $$(T^{4} + 13277 T^{2} + 13498276)^{2}$$
$19$ $$(T^{2} + 35 T - 4850)^{4}$$
$23$ $$T^{8} - 44373 T^{6} + \cdots + 32\!\cdots\!96$$
$29$ $$(T^{4} - 325 T^{3} + 91767 T^{2} + \cdots + 192044164)^{2}$$
$31$ $$(T^{2} - 6 T + 36)^{4}$$
$37$ $$(T^{4} + 205172 T^{2} + \cdots + 10188076096)^{2}$$
$41$ $$(T^{4} + 238 T^{3} + 72183 T^{2} + \cdots + 241460521)^{2}$$
$43$ $$T^{8} - 129593 T^{6} + \cdots + 13\!\cdots\!96$$
$47$ $$T^{8} - 407897 T^{6} + \cdots + 16\!\cdots\!16$$
$53$ $$(T^{4} + 190088 T^{2} + \cdots + 4893841936)^{2}$$
$59$ $$(T^{4} + 85 T^{3} + 25227 T^{2} + \cdots + 324072004)^{2}$$
$61$ $$(T^{4} - 247 T^{3} + \cdots + 89074790116)^{2}$$
$67$ $$T^{8} - 742242 T^{6} + \cdots + 12\!\cdots\!81$$
$71$ $$(T^{2} - 394 T - 61016)^{4}$$
$73$ $$(T^{4} + 1022273 T^{2} + \cdots + 33224540176)^{2}$$
$79$ $$(T^{4} + 840 T^{3} + \cdots + 16576047504)^{2}$$
$83$ $$T^{8} - 3460833 T^{6} + \cdots + 75\!\cdots\!76$$
$89$ $$(T^{2} - 1065 T - 535050)^{4}$$
$97$ $$T^{8} - 814637 T^{6} + \cdots + 23\!\cdots\!36$$