# Properties

 Label 1728.2.r.f Level $1728$ Weight $2$ Character orbit 1728.r Analytic conductor $13.798$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36$$ x^12 - 16*x^8 - 24*x^7 + 96*x^5 + 304*x^4 + 384*x^3 + 288*x^2 + 144*x + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 576) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} + 2) q^{5} + (\beta_{11} + \beta_{2}) q^{7}+O(q^{10})$$ q + (b4 + 2) * q^5 + (b11 + b2) * q^7 $$q + (\beta_{4} + 2) q^{5} + (\beta_{11} + \beta_{2}) q^{7} + ( - \beta_{11} - \beta_{9} - \beta_{5}) q^{11} - \beta_{10} q^{13} + \beta_1 q^{17} - \beta_{7} q^{19} + ( - 2 \beta_{11} - 2 \beta_{9} - 2 \beta_{7} - \beta_{5} + \beta_{3} - \beta_{2}) q^{23} + ( - 2 \beta_{4} - 2) q^{25} + ( - \beta_{6} + \beta_{4} + \beta_1 - 1) q^{29} + (2 \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2}) q^{31} + (2 \beta_{11} + \beta_{9} + \beta_{5} + 2 \beta_{2}) q^{35} + ( - \beta_{10} + \beta_{8} + 2 \beta_{4} + 2) q^{37} + (\beta_{6} + 3 \beta_{4}) q^{41} + (\beta_{11} + 2 \beta_{9} + 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3} + \beta_{2}) q^{43} + (\beta_{11} - 2 \beta_{9} + \beta_{7} + 2 \beta_{5} + \beta_{3} - \beta_{2}) q^{47} + ( - \beta_{10} + 2 \beta_{8} + 5 \beta_{4} + 2) q^{49} + ( - 2 \beta_{6} - 4 \beta_{4} - \beta_1 - 2) q^{53} + ( - \beta_{11} - \beta_{9} - 2 \beta_{5} + \beta_{2}) q^{55} + ( - \beta_{11} + \beta_{9} - \beta_{5} + 3 \beta_{3} + \beta_{2}) q^{59} + ( - \beta_{6} - \beta_{4} + \beta_1 + 1) q^{61} + ( - 2 \beta_{10} + \beta_{8} + 1) q^{65} + ( - 2 \beta_{11} + 2 \beta_{9} - \beta_{5} + 2 \beta_{3} + \beta_{2}) q^{67} + (\beta_{11} - 2 \beta_{9} + 2 \beta_{7} - \beta_{5} - 4 \beta_{3} - \beta_{2}) q^{71} + (\beta_{10} + \beta_{8}) q^{73} + ( - 3 \beta_{10} - \beta_{6} + 6 \beta_{4} - 2 \beta_1 + 12) q^{77} + (\beta_{11} - \beta_{9} - \beta_{7} + \beta_{5} - \beta_{3}) q^{79} + (\beta_{11} + 2 \beta_{9} + 3 \beta_{7} + 2 \beta_{5} - 3 \beta_{3} + \beta_{2}) q^{83} + (\beta_{6} + 2 \beta_1) q^{85} + (\beta_{10} + \beta_{8} + 4) q^{89} + (3 \beta_{11} + 3 \beta_{9} + 3 \beta_{2}) q^{91} + ( - 2 \beta_{7} + \beta_{3}) q^{95} + (3 \beta_{6} - \beta_{4} + 3 \beta_1 - 1) q^{97}+O(q^{100})$$ q + (b4 + 2) * q^5 + (b11 + b2) * q^7 + (-b11 - b9 - b5) * q^11 - b10 * q^13 + b1 * q^17 - b7 * q^19 + (-2*b11 - 2*b9 - 2*b7 - b5 + b3 - b2) * q^23 + (-2*b4 - 2) * q^25 + (-b6 + b4 + b1 - 1) * q^29 + (2*b7 + b5 - b3 + b2) * q^31 + (2*b11 + b9 + b5 + 2*b2) * q^35 + (-b10 + b8 + 2*b4 + 2) * q^37 + (b6 + 3*b4) * q^41 + (b11 + 2*b9 + 2*b7 + 2*b5 - 2*b3 + b2) * q^43 + (b11 - 2*b9 + b7 + 2*b5 + b3 - b2) * q^47 + (-b10 + 2*b8 + 5*b4 + 2) * q^49 + (-2*b6 - 4*b4 - b1 - 2) * q^53 + (-b11 - b9 - 2*b5 + b2) * q^55 + (-b11 + b9 - b5 + 3*b3 + b2) * q^59 + (-b6 - b4 + b1 + 1) * q^61 + (-2*b10 + b8 + 1) * q^65 + (-2*b11 + 2*b9 - b5 + 2*b3 + b2) * q^67 + (b11 - 2*b9 + 2*b7 - b5 - 4*b3 - b2) * q^71 + (b10 + b8) * q^73 + (-3*b10 - b6 + 6*b4 - 2*b1 + 12) * q^77 + (b11 - b9 - b7 + b5 - b3) * q^79 + (b11 + 2*b9 + 3*b7 + 2*b5 - 3*b3 + b2) * q^83 + (b6 + 2*b1) * q^85 + (b10 + b8 + 4) * q^89 + (3*b11 + 3*b9 + 3*b2) * q^91 + (-2*b7 + b3) * q^95 + (3*b6 - b4 + 3*b1 - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 18 q^{5}+O(q^{10})$$ 12 * q + 18 * q^5 $$12 q + 18 q^{5} - 6 q^{13} - 12 q^{25} - 18 q^{29} - 18 q^{41} - 24 q^{49} + 18 q^{61} - 6 q^{65} + 90 q^{77} + 48 q^{89} - 6 q^{97}+O(q^{100})$$ 12 * q + 18 * q^5 - 6 * q^13 - 12 * q^25 - 18 * q^29 - 18 * q^41 - 24 * q^49 + 18 * q^61 - 6 * q^65 + 90 * q^77 + 48 * q^89 - 6 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36$$ :

 $$\beta_{1}$$ $$=$$ $$( - 275 \nu^{11} + 300 \nu^{10} - 650 \nu^{9} + 1148 \nu^{8} + 1747 \nu^{7} + 3300 \nu^{6} + 5062 \nu^{5} - 27262 \nu^{4} - 51252 \nu^{3} - 44964 \nu^{2} - 26082 \nu - 96192 ) / 51972$$ (-275*v^11 + 300*v^10 - 650*v^9 + 1148*v^8 + 1747*v^7 + 3300*v^6 + 5062*v^5 - 27262*v^4 - 51252*v^3 - 44964*v^2 - 26082*v - 96192) / 51972 $$\beta_{2}$$ $$=$$ $$( 839 \nu^{11} + 7708 \nu^{10} - 10784 \nu^{9} + 8976 \nu^{8} - 22648 \nu^{7} - 131336 \nu^{6} - 31862 \nu^{5} + 209952 \nu^{4} + 921960 \nu^{3} + 1654212 \nu^{2} + \cdots + 275832 ) / 103944$$ (839*v^11 + 7708*v^10 - 10784*v^9 + 8976*v^8 - 22648*v^7 - 131336*v^6 - 31862*v^5 + 209952*v^4 + 921960*v^3 + 1654212*v^2 + 708624*v + 275832) / 103944 $$\beta_{3}$$ $$=$$ $$( 962 \nu^{11} - 2392 \nu^{10} + 2887 \nu^{9} - 2220 \nu^{8} - 13990 \nu^{7} + 14442 \nu^{6} + 11380 \nu^{5} + 57708 \nu^{4} + 98118 \nu^{3} - 132080 \nu^{2} + 49284 \nu + 15984 ) / 51972$$ (962*v^11 - 2392*v^10 + 2887*v^9 - 2220*v^8 - 13990*v^7 + 14442*v^6 + 11380*v^5 + 57708*v^4 + 98118*v^3 - 132080*v^2 + 49284*v + 15984) / 51972 $$\beta_{4}$$ $$=$$ $$( - 118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} - 10408 \nu^{4} - 27758 \nu^{3} - 22776 \nu^{2} - 12468 \nu - 5172 ) / 1704$$ (-118*v^11 + 90*v^10 - 53*v^9 + 32*v^8 + 1866*v^7 + 1416*v^6 - 1364*v^5 - 10408*v^4 - 27758*v^3 - 22776*v^2 - 12468*v - 5172) / 1704 $$\beta_{5}$$ $$=$$ $$( 2793 \nu^{11} - 3602 \nu^{10} + 4065 \nu^{9} - 2808 \nu^{8} - 44568 \nu^{7} - 7530 \nu^{6} + 17238 \nu^{5} + 219348 \nu^{4} + 562962 \nu^{3} + 348980 \nu^{2} + \cdots + 133704 ) / 34648$$ (2793*v^11 - 3602*v^10 + 4065*v^9 - 2808*v^8 - 44568*v^7 - 7530*v^6 + 17238*v^5 + 219348*v^4 + 562962*v^3 + 348980*v^2 + 364968*v + 133704) / 34648 $$\beta_{6}$$ $$=$$ $$( 6049 \nu^{11} - 4740 \nu^{10} + 3454 \nu^{9} - 2632 \nu^{8} - 93803 \nu^{7} - 72588 \nu^{6} + 65542 \nu^{5} + 530594 \nu^{4} + 1432764 \nu^{3} + 1177668 \nu^{2} + \cdots + 268416 ) / 51972$$ (6049*v^11 - 4740*v^10 + 3454*v^9 - 2632*v^8 - 93803*v^7 - 72588*v^6 + 65542*v^5 + 530594*v^4 + 1432764*v^3 + 1177668*v^2 + 645714*v + 268416) / 51972 $$\beta_{7}$$ $$=$$ $$( 1688 \nu^{11} - 1054 \nu^{10} + 840 \nu^{9} - 684 \nu^{8} - 26378 \nu^{7} - 24587 \nu^{6} + 12648 \nu^{5} + 151968 \nu^{4} + 420176 \nu^{3} + 400856 \nu^{2} + 282372 \nu + 105120 ) / 12993$$ (1688*v^11 - 1054*v^10 + 840*v^9 - 684*v^8 - 26378*v^7 - 24587*v^6 + 12648*v^5 + 151968*v^4 + 420176*v^3 + 400856*v^2 + 282372*v + 105120) / 12993 $$\beta_{8}$$ $$=$$ $$( - 15676 \nu^{11} + 10866 \nu^{10} - 681 \nu^{9} - 11340 \nu^{8} + 267718 \nu^{7} + 188112 \nu^{6} - 234336 \nu^{5} - 1340808 \nu^{4} - 3664246 \nu^{3} + \cdots - 1037028 ) / 103944$$ (-15676*v^11 + 10866*v^10 - 681*v^9 - 11340*v^8 + 267718*v^7 + 188112*v^6 - 234336*v^5 - 1340808*v^4 - 3664246*v^3 - 2984520*v^2 - 1622652*v - 1037028) / 103944 $$\beta_{9}$$ $$=$$ $$( - 19630 \nu^{11} + 7150 \nu^{10} - 3611 \nu^{9} + 1848 \nu^{8} + 317744 \nu^{7} + 348166 \nu^{6} - 103124 \nu^{5} - 1841148 \nu^{4} - 5346606 \nu^{3} + \cdots - 1369008 ) / 103944$$ (-19630*v^11 + 7150*v^10 - 3611*v^9 + 1848*v^8 + 317744*v^7 + 348166*v^6 - 103124*v^5 - 1841148*v^4 - 5346606*v^3 - 5685648*v^2 - 3656232*v - 1369008) / 103944 $$\beta_{10}$$ $$=$$ $$( 284 \nu^{11} - 222 \nu^{10} + 159 \nu^{9} - 192 \nu^{8} - 4304 \nu^{7} - 3408 \nu^{6} + 2664 \nu^{5} + 26820 \nu^{4} + 65498 \nu^{3} + 53952 \nu^{2} + 29640 \nu + 4860 ) / 1464$$ (284*v^11 - 222*v^10 + 159*v^9 - 192*v^8 - 4304*v^7 - 3408*v^6 + 2664*v^5 + 26820*v^4 + 65498*v^3 + 53952*v^2 + 29640*v + 4860) / 1464 $$\beta_{11}$$ $$=$$ $$( - 6916 \nu^{11} + 3388 \nu^{10} - 2205 \nu^{9} + 1476 \nu^{8} + 110866 \nu^{7} + 108978 \nu^{6} - 40656 \nu^{5} - 637476 \nu^{4} - 1820914 \nu^{3} - 1805344 \nu^{2} + \cdots - 462528 ) / 34648$$ (-6916*v^11 + 3388*v^10 - 2205*v^9 + 1476*v^8 + 110866*v^7 + 108978*v^6 - 40656*v^5 - 637476*v^4 - 1820914*v^3 - 1805344*v^2 - 1237836*v - 462528) / 34648
 $$\nu$$ $$=$$ $$( -2\beta_{11} + \beta_{9} - \beta_{6} - \beta_{5} - \beta_{2} + \beta_1 ) / 6$$ (-2*b11 + b9 - b6 - b5 - b2 + b1) / 6 $$\nu^{2}$$ $$=$$ $$( -2\beta_{11} + 2\beta_{9} + 2\beta_{5} - 9\beta_{3} - 2\beta_{2} ) / 6$$ (-2*b11 + 2*b9 + 2*b5 - 9*b3 - 2*b2) / 6 $$\nu^{3}$$ $$=$$ $$( - 4 \beta_{11} + 8 \beta_{9} + 3 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} + 12 \beta_{4} - 6 \beta_{3} + 4 \beta_{2} + 4 \beta _1 + 6 ) / 6$$ (-4*b11 + 8*b9 + 3*b7 + 8*b6 + 4*b5 + 12*b4 - 6*b3 + 4*b2 + 4*b1 + 6) / 6 $$\nu^{4}$$ $$=$$ $$( 4\beta_{10} - 2\beta_{8} + 9\beta_{6} + 30\beta_{4} + 9\beta _1 + 28 ) / 3$$ (4*b10 - 2*b8 + 9*b6 + 30*b4 + 9*b1 + 28) / 3 $$\nu^{5}$$ $$=$$ $$( - 22 \beta_{11} + 3 \beta_{10} - 22 \beta_{9} - 42 \beta_{7} + 19 \beta_{6} - 38 \beta_{5} + 39 \beta_{4} + 21 \beta_{3} - 38 \beta_{2} + 38 \beta _1 + 78 ) / 6$$ (-22*b11 + 3*b10 - 22*b9 - 42*b7 + 19*b6 - 38*b5 + 39*b4 + 21*b3 - 38*b2 + 38*b1 + 78) / 6 $$\nu^{6}$$ $$=$$ $$( -56\beta_{11} - 16\beta_{9} - 81\beta_{7} - 40\beta_{5} - 56\beta_{2} ) / 3$$ (-56*b11 - 16*b9 - 81*b7 - 40*b5 - 56*b2) / 3 $$\nu^{7}$$ $$=$$ $$( - 94 \beta_{11} + 35 \beta_{9} - 12 \beta_{8} - 60 \beta_{7} + 47 \beta_{6} - 35 \beta_{5} + 108 \beta_{4} - 60 \beta_{3} - 59 \beta_{2} - 47 \beta _1 - 120 ) / 3$$ (-94*b11 + 35*b9 - 12*b8 - 60*b7 + 47*b6 - 35*b5 + 108*b4 - 60*b3 - 59*b2 - 47*b1 - 120) / 3 $$\nu^{8}$$ $$=$$ $$( 35\beta_{10} - 70\beta_{8} + 249\beta_{6} + 669\beta_{4} - 70 ) / 3$$ (35*b10 - 70*b8 + 249*b6 + 669*b4 - 70) / 3 $$\nu^{9}$$ $$=$$ $$( 164 \beta_{11} + 72 \beta_{10} - 472 \beta_{9} - 72 \beta_{8} - 321 \beta_{7} + 472 \beta_{6} - 308 \beta_{5} + 1140 \beta_{4} + 642 \beta_{3} - 164 \beta_{2} + 236 \beta _1 + 498 ) / 3$$ (164*b11 + 72*b10 - 472*b9 - 72*b8 - 321*b7 + 472*b6 - 308*b5 + 1140*b4 + 642*b3 - 164*b2 + 236*b1 + 498) / 3 $$\nu^{10}$$ $$=$$ $$( -328\beta_{11} - 1442\beta_{9} - 1908\beta_{7} - 1442\beta_{5} + 1908\beta_{3} - 1114\beta_{2} ) / 3$$ (-328*b11 - 1442*b9 - 1908*b7 - 1442*b5 + 1908*b3 - 1114*b2) / 3 $$\nu^{11}$$ $$=$$ $$( - 1586 \beta_{11} - 393 \beta_{10} - 1586 \beta_{9} - 3342 \beta_{7} - 1193 \beta_{6} - 2386 \beta_{5} - 2949 \beta_{4} + 1671 \beta_{3} - 2386 \beta_{2} - 2386 \beta _1 - 5898 ) / 3$$ (-1586*b11 - 393*b10 - 1586*b9 - 3342*b7 - 1193*b6 - 2386*b5 - 2949*b4 + 1671*b3 - 2386*b2 - 2386*b1 - 5898) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −0.180407 − 0.673288i 2.17840 − 0.583700i −0.403293 − 1.50511i −1.50511 + 0.403293i 0.583700 + 2.17840i −0.673288 + 0.180407i −0.180407 + 0.673288i 2.17840 + 0.583700i −0.403293 + 1.50511i −1.50511 − 0.403293i 0.583700 − 2.17840i −0.673288 − 0.180407i
0 0 0 1.50000 0.866025i 0 −2.17731 + 3.77121i 0 0 0
289.2 0 0 0 1.50000 0.866025i 0 −1.80664 + 3.12920i 0 0 0
289.3 0 0 0 1.50000 0.866025i 0 −0.495361 + 0.857990i 0 0 0
289.4 0 0 0 1.50000 0.866025i 0 0.495361 0.857990i 0 0 0
289.5 0 0 0 1.50000 0.866025i 0 1.80664 3.12920i 0 0 0
289.6 0 0 0 1.50000 0.866025i 0 2.17731 3.77121i 0 0 0
1441.1 0 0 0 1.50000 + 0.866025i 0 −2.17731 3.77121i 0 0 0
1441.2 0 0 0 1.50000 + 0.866025i 0 −1.80664 3.12920i 0 0 0
1441.3 0 0 0 1.50000 + 0.866025i 0 −0.495361 0.857990i 0 0 0
1441.4 0 0 0 1.50000 + 0.866025i 0 0.495361 + 0.857990i 0 0 0
1441.5 0 0 0 1.50000 + 0.866025i 0 1.80664 + 3.12920i 0 0 0
1441.6 0 0 0 1.50000 + 0.866025i 0 2.17731 + 3.77121i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1441.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
72.n even 6 1 inner
72.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.r.f 12
3.b odd 2 1 576.2.r.e 12
4.b odd 2 1 inner 1728.2.r.f 12
8.b even 2 1 1728.2.r.e 12
8.d odd 2 1 1728.2.r.e 12
9.c even 3 1 1728.2.r.e 12
9.c even 3 1 5184.2.d.r 12
9.d odd 6 1 576.2.r.f yes 12
9.d odd 6 1 5184.2.d.q 12
12.b even 2 1 576.2.r.e 12
24.f even 2 1 576.2.r.f yes 12
24.h odd 2 1 576.2.r.f yes 12
36.f odd 6 1 1728.2.r.e 12
36.f odd 6 1 5184.2.d.r 12
36.h even 6 1 576.2.r.f yes 12
36.h even 6 1 5184.2.d.q 12
72.j odd 6 1 576.2.r.e 12
72.j odd 6 1 5184.2.d.q 12
72.l even 6 1 576.2.r.e 12
72.l even 6 1 5184.2.d.q 12
72.n even 6 1 inner 1728.2.r.f 12
72.n even 6 1 5184.2.d.r 12
72.p odd 6 1 inner 1728.2.r.f 12
72.p odd 6 1 5184.2.d.r 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.e 12 3.b odd 2 1
576.2.r.e 12 12.b even 2 1
576.2.r.e 12 72.j odd 6 1
576.2.r.e 12 72.l even 6 1
576.2.r.f yes 12 9.d odd 6 1
576.2.r.f yes 12 24.f even 2 1
576.2.r.f yes 12 24.h odd 2 1
576.2.r.f yes 12 36.h even 6 1
1728.2.r.e 12 8.b even 2 1
1728.2.r.e 12 8.d odd 2 1
1728.2.r.e 12 9.c even 3 1
1728.2.r.e 12 36.f odd 6 1
1728.2.r.f 12 1.a even 1 1 trivial
1728.2.r.f 12 4.b odd 2 1 inner
1728.2.r.f 12 72.n even 6 1 inner
1728.2.r.f 12 72.p odd 6 1 inner
5184.2.d.q 12 9.d odd 6 1
5184.2.d.q 12 36.h even 6 1
5184.2.d.q 12 72.j odd 6 1
5184.2.d.q 12 72.l even 6 1
5184.2.d.r 12 9.c even 3 1
5184.2.d.r 12 36.f odd 6 1
5184.2.d.r 12 72.n even 6 1
5184.2.d.r 12 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 3T_{5} + 3$$ acting on $$S_{2}^{\mathrm{new}}(1728, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$(T^{2} - 3 T + 3)^{6}$$
$7$ $$T^{12} + 33 T^{10} + 810 T^{8} + \cdots + 59049$$
$11$ $$T^{12} - 39 T^{10} + 1350 T^{8} + \cdots + 6561$$
$13$ $$(T^{6} + 3 T^{5} - 24 T^{4} - 81 T^{3} + \cdots + 243)^{2}$$
$17$ $$(T^{3} - 24 T - 36)^{4}$$
$19$ $$(T^{2} + 4)^{6}$$
$23$ $$T^{12} + 117 T^{10} + \cdots + 2492305929$$
$29$ $$(T^{6} + 9 T^{5} - 36 T^{4} - 567 T^{3} + \cdots + 93987)^{2}$$
$31$ $$T^{12} + 117 T^{10} + \cdots + 95004009$$
$37$ $$(T^{6} + 60 T^{4} + 1008 T^{2} + \cdots + 3888)^{2}$$
$41$ $$(T^{6} + 9 T^{5} + 78 T^{4} + 45 T^{3} + \cdots + 81)^{2}$$
$43$ $$T^{12} - 123 T^{10} + \cdots + 47458321$$
$47$ $$T^{12} + 297 T^{10} + \cdots + 514609673769$$
$53$ $$(T^{6} + 180 T^{4} + 1728 T^{2} + \cdots + 432)^{2}$$
$59$ $$T^{12} - 147 T^{10} + 20898 T^{8} + \cdots + 531441$$
$61$ $$(T^{6} - 9 T^{5} - 36 T^{4} + 567 T^{3} + \cdots + 4563)^{2}$$
$67$ $$T^{12} - 231 T^{10} + \cdots + 352275361$$
$71$ $$(T^{6} - 288 T^{4} + 19872 T^{2} + \cdots - 124848)^{2}$$
$73$ $$(T^{3} - 84 T - 164)^{4}$$
$79$ $$T^{12} + 93 T^{10} + 7758 T^{8} + \cdots + 4782969$$
$83$ $$T^{12} - 171 T^{10} + \cdots + 43046721$$
$89$ $$(T^{3} - 12 T^{2} - 36 T + 108)^{4}$$
$97$ $$(T^{6} + 3 T^{5} + 222 T^{4} + \cdots + 1408969)^{2}$$