# Properties

 Label 1728.3.b.j Level $1728$ Weight $3$ Character orbit 1728.b Analytic conductor $47.085$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(1567,1728)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1728.1567");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.116304318664704.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 2x^{11} + x^{10} + 6x^{9} - 9x^{8} - 2x^{7} + 18x^{6} - 4x^{5} - 36x^{4} + 48x^{3} + 16x^{2} - 64x + 64$$ x^12 - 2*x^11 + x^10 + 6*x^9 - 9*x^8 - 2*x^7 + 18*x^6 - 4*x^5 - 36*x^4 + 48*x^3 + 16*x^2 - 64*x + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{22}\cdot 3^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{5} - \beta_{2} q^{7}+O(q^{10})$$ q - b4 * q^5 - b2 * q^7 $$q - \beta_{4} q^{5} - \beta_{2} q^{7} + ( - \beta_{9} + 2 \beta_1) q^{11} + (\beta_{5} + 2 \beta_{4}) q^{13} + (\beta_{10} + 4) q^{17} + ( - \beta_{11} - \beta_{9} + 4 \beta_1) q^{19} + (\beta_{7} + 3 \beta_{3} + 3 \beta_{2}) q^{23} + ( - \beta_{10} - 2 \beta_{8} - 6) q^{25} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{4}) q^{29} + ( - 3 \beta_{7} + 4 \beta_{2}) q^{31} + (2 \beta_{9} + 5 \beta_1) q^{35} + (\beta_{6} - 3 \beta_{5} + 6 \beta_{4}) q^{37} + ( - 2 \beta_{10} - 3 \beta_{8} + 28) q^{41} + ( - \beta_{11} + 2 \beta_{9} - 9 \beta_1) q^{43} + ( - 4 \beta_{7} + 9 \beta_{3}) q^{47} + (\beta_{10} + 2 \beta_{8} + 4) q^{49} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4}) q^{53} + ( - 2 \beta_{3} - 5 \beta_{2}) q^{55} + ( - 2 \beta_{11} - 2 \beta_{9} + 24 \beta_1) q^{59} - 4 \beta_{6} q^{61} + (\beta_{10} + 6 \beta_{8} + 76) q^{65} + (\beta_{11} + 4 \beta_{9} - \beta_1) q^{67} + (\beta_{7} + 18 \beta_{3} - 9 \beta_{2}) q^{71} + (\beta_{10} - 4 \beta_{8} + 5) q^{73} + ( - 10 \beta_{6} - 4 \beta_{5} - 7 \beta_{4}) q^{77} + (3 \beta_{7} + 8 \beta_{3} + 3 \beta_{2}) q^{79} + (4 \beta_{11} + 5 \beta_{9} + 22 \beta_1) q^{83} + (2 \beta_{6} + 5 \beta_{5} - 14 \beta_{4}) q^{85} + ( - \beta_{10} + 3 \beta_{8} + 104) q^{89} + (2 \beta_{11} - 7 \beta_{9} + 31 \beta_1) q^{91} + (13 \beta_{7} + 27 \beta_{3} - 9 \beta_{2}) q^{95} + (2 \beta_{10} + 10 \beta_{8} - 17) q^{97}+O(q^{100})$$ q - b4 * q^5 - b2 * q^7 + (-b9 + 2*b1) * q^11 + (b5 + 2*b4) * q^13 + (b10 + 4) * q^17 + (-b11 - b9 + 4*b1) * q^19 + (b7 + 3*b3 + 3*b2) * q^23 + (-b10 - 2*b8 - 6) * q^25 + (-2*b6 + b5 + 2*b4) * q^29 + (-3*b7 + 4*b2) * q^31 + (2*b9 + 5*b1) * q^35 + (b6 - 3*b5 + 6*b4) * q^37 + (-2*b10 - 3*b8 + 28) * q^41 + (-b11 + 2*b9 - 9*b1) * q^43 + (-4*b7 + 9*b3) * q^47 + (b10 + 2*b8 + 4) * q^49 + (-2*b6 + b5 + b4) * q^53 + (-2*b3 - 5*b2) * q^55 + (-2*b11 - 2*b9 + 24*b1) * q^59 - 4*b6 * q^61 + (b10 + 6*b8 + 76) * q^65 + (b11 + 4*b9 - b1) * q^67 + (b7 + 18*b3 - 9*b2) * q^71 + (b10 - 4*b8 + 5) * q^73 + (-10*b6 - 4*b5 - 7*b4) * q^77 + (3*b7 + 8*b3 + 3*b2) * q^79 + (4*b11 + 5*b9 + 22*b1) * q^83 + (2*b6 + 5*b5 - 14*b4) * q^85 + (-b10 + 3*b8 + 104) * q^89 + (2*b11 - 7*b9 + 31*b1) * q^91 + (13*b7 + 27*b3 - 9*b2) * q^95 + (2*b10 + 10*b8 - 17) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q+O(q^{10})$$ 12 * q $$12 q + 48 q^{17} - 72 q^{25} + 336 q^{41} + 48 q^{49} + 912 q^{65} + 60 q^{73} + 1248 q^{89} - 204 q^{97}+O(q^{100})$$ 12 * q + 48 * q^17 - 72 * q^25 + 336 * q^41 + 48 * q^49 + 912 * q^65 + 60 * q^73 + 1248 * q^89 - 204 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2x^{11} + x^{10} + 6x^{9} - 9x^{8} - 2x^{7} + 18x^{6} - 4x^{5} - 36x^{4} + 48x^{3} + 16x^{2} - 64x + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( - 3 \nu^{11} - 2 \nu^{10} + 9 \nu^{9} - 10 \nu^{8} - 25 \nu^{7} + 30 \nu^{6} + 30 \nu^{5} - 68 \nu^{4} - 28 \nu^{3} + 112 \nu^{2} - 64 \nu - 128 ) / 32$$ (-3*v^11 - 2*v^10 + 9*v^9 - 10*v^8 - 25*v^7 + 30*v^6 + 30*v^5 - 68*v^4 - 28*v^3 + 112*v^2 - 64*v - 128) / 32 $$\beta_{2}$$ $$=$$ $$( - \nu^{11} - 8 \nu^{10} + 7 \nu^{9} + 4 \nu^{8} - 35 \nu^{7} + 16 \nu^{6} + 66 \nu^{5} - 52 \nu^{4} - 96 \nu^{3} + 144 \nu^{2} - 32 \nu - 248 ) / 8$$ (-v^11 - 8*v^10 + 7*v^9 + 4*v^8 - 35*v^7 + 16*v^6 + 66*v^5 - 52*v^4 - 96*v^3 + 144*v^2 - 32*v - 248) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{11} - 10 \nu^{10} + 5 \nu^{9} + 14 \nu^{8} - 37 \nu^{7} - 10 \nu^{6} + 78 \nu^{5} - 20 \nu^{4} - 156 \nu^{3} + 144 \nu^{2} + 32 \nu - 256 ) / 8$$ (v^11 - 10*v^10 + 5*v^9 + 14*v^8 - 37*v^7 - 10*v^6 + 78*v^5 - 20*v^4 - 156*v^3 + 144*v^2 + 32*v - 256) / 8 $$\beta_{4}$$ $$=$$ $$( 7 \nu^{11} - 30 \nu^{10} - 21 \nu^{9} + 122 \nu^{8} - 139 \nu^{7} - 158 \nu^{6} + 330 \nu^{5} + 148 \nu^{4} - 772 \nu^{3} + 208 \nu^{2} + 960 \nu - 1408 ) / 32$$ (7*v^11 - 30*v^10 - 21*v^9 + 122*v^8 - 139*v^7 - 158*v^6 + 330*v^5 + 148*v^4 - 772*v^3 + 208*v^2 + 960*v - 1408) / 32 $$\beta_{5}$$ $$=$$ $$( - 2 \nu^{11} + 11 \nu^{10} - 2 \nu^{9} - 23 \nu^{8} + 38 \nu^{7} + 27 \nu^{6} - 84 \nu^{5} - 16 \nu^{4} + 180 \nu^{3} - 96 \nu^{2} - 96 \nu + 256 ) / 4$$ (-2*v^11 + 11*v^10 - 2*v^9 - 23*v^8 + 38*v^7 + 27*v^6 - 84*v^5 - 16*v^4 + 180*v^3 - 96*v^2 - 96*v + 256) / 4 $$\beta_{6}$$ $$=$$ $$( - \nu^{11} + 5 \nu^{10} + \nu^{9} - 15 \nu^{8} + 19 \nu^{7} + 19 \nu^{6} - 48 \nu^{5} - 18 \nu^{4} + 104 \nu^{3} - 44 \nu^{2} - 96 \nu + 168 ) / 2$$ (-v^11 + 5*v^10 + v^9 - 15*v^8 + 19*v^7 + 19*v^6 - 48*v^5 - 18*v^4 + 104*v^3 - 44*v^2 - 96*v + 168) / 2 $$\beta_{7}$$ $$=$$ $$( 9 \nu^{11} - 30 \nu^{10} + 9 \nu^{9} + 66 \nu^{8} - 93 \nu^{7} - 78 \nu^{6} + 222 \nu^{5} + 60 \nu^{4} - 480 \nu^{3} + 336 \nu^{2} + 288 \nu - 552 ) / 8$$ (9*v^11 - 30*v^10 + 9*v^9 + 66*v^8 - 93*v^7 - 78*v^6 + 222*v^5 + 60*v^4 - 480*v^3 + 336*v^2 + 288*v - 552) / 8 $$\beta_{8}$$ $$=$$ $$( 15 \nu^{11} - 12 \nu^{10} - 21 \nu^{9} + 72 \nu^{8} - 3 \nu^{7} - 132 \nu^{6} + 66 \nu^{5} + 252 \nu^{4} - 300 \nu^{3} - 48 \nu^{2} + 480 \nu - 144 ) / 8$$ (15*v^11 - 12*v^10 - 21*v^9 + 72*v^8 - 3*v^7 - 132*v^6 + 66*v^5 + 252*v^4 - 300*v^3 - 48*v^2 + 480*v - 144) / 8 $$\beta_{9}$$ $$=$$ $$( - 67 \nu^{11} + 86 \nu^{10} + 73 \nu^{9} - 386 \nu^{8} + 231 \nu^{7} + 630 \nu^{6} - 770 \nu^{5} - 900 \nu^{4} + 2020 \nu^{3} - 528 \nu^{2} - 2624 \nu + 2176 ) / 32$$ (-67*v^11 + 86*v^10 + 73*v^9 - 386*v^8 + 231*v^7 + 630*v^6 - 770*v^5 - 900*v^4 + 2020*v^3 - 528*v^2 - 2624*v + 2176) / 32 $$\beta_{10}$$ $$=$$ $$( 18 \nu^{11} - 15 \nu^{10} - 30 \nu^{9} + 87 \nu^{8} - 6 \nu^{7} - 171 \nu^{6} + 84 \nu^{5} + 300 \nu^{4} - 324 \nu^{3} - 48 \nu^{2} + 576 \nu - 184 ) / 4$$ (18*v^11 - 15*v^10 - 30*v^9 + 87*v^8 - 6*v^7 - 171*v^6 + 84*v^5 + 300*v^4 - 324*v^3 - 48*v^2 + 576*v - 184) / 4 $$\beta_{11}$$ $$=$$ $$( 157 \nu^{11} - 146 \nu^{10} - 247 \nu^{9} + 806 \nu^{8} - 153 \nu^{7} - 1554 \nu^{6} + 1022 \nu^{5} + 2652 \nu^{4} - 3484 \nu^{3} - 336 \nu^{2} + 6080 \nu - 2560 ) / 32$$ (157*v^11 - 146*v^10 - 247*v^9 + 806*v^8 - 153*v^7 - 1554*v^6 + 1022*v^5 + 2652*v^4 - 3484*v^3 - 336*v^2 + 6080*v - 2560) / 32
 $$\nu$$ $$=$$ $$( 2 \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - \beta_{3} + 2 \beta_{2} + 8 ) / 48$$ (2*b11 - b10 + 2*b9 - b8 + 2*b7 + 2*b6 + 2*b5 + 4*b4 - b3 + 2*b2 + 8) / 48 $$\nu^{2}$$ $$=$$ $$( - \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 3 \beta_{6} + 6 \beta_{5} + 4 \beta_{3} - 2 \beta_{2} + 15 \beta _1 + 8 ) / 48$$ (-b11 + 2*b10 + 2*b9 + 2*b7 - 3*b6 + 6*b5 + 4*b3 - 2*b2 + 15*b1 + 8) / 48 $$\nu^{3}$$ $$=$$ $$( \beta_{10} - 3\beta_{8} + 2\beta_{7} - 7\beta_{3} + 2\beta_{2} - 32 ) / 24$$ (b10 - 3*b8 + 2*b7 - 7*b3 + 2*b2 - 32) / 24 $$\nu^{4}$$ $$=$$ $$( - \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - 11 \beta_{6} - 2 \beta_{5} - 16 \beta_{4} - 16 \beta_{3} + 2 \beta_{2} + 63 \beta _1 - 40 ) / 48$$ (-b11 + 2*b10 + 2*b9 + 4*b8 - 2*b7 - 11*b6 - 2*b5 - 16*b4 - 16*b3 + 2*b2 + 63*b1 - 40) / 48 $$\nu^{5}$$ $$=$$ $$( 2 \beta_{11} - 3 \beta_{10} - 10 \beta_{9} - 7 \beta_{8} - 6 \beta_{7} - 14 \beta_{6} - 2 \beta_{5} - 28 \beta_{4} - 11 \beta_{3} + 10 \beta_{2} - 36 \beta _1 + 48 ) / 48$$ (2*b11 - 3*b10 - 10*b9 - 7*b8 - 6*b7 - 14*b6 - 2*b5 - 28*b4 - 11*b3 + 10*b2 - 36*b1 + 48) / 48 $$\nu^{6}$$ $$=$$ $$( -\beta_{10} + 4\beta_{8} - \beta_{7} - 6\beta_{3} + 9\beta_{2} + 44 ) / 12$$ (-b10 + 4*b8 - b7 - 6*b3 + 9*b2 + 44) / 12 $$\nu^{7}$$ $$=$$ $$( 2 \beta_{11} - 3 \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{7} - 6 \beta_{6} + 10 \beta_{5} - 4 \beta_{4} + 13 \beta_{3} + 2 \beta_{2} - 56 \beta _1 - 16 ) / 16$$ (2*b11 - 3*b10 + 2*b9 + b8 + 2*b7 - 6*b6 + 10*b5 - 4*b4 + 13*b3 + 2*b2 - 56*b1 - 16) / 16 $$\nu^{8}$$ $$=$$ $$( - 23 \beta_{11} + 10 \beta_{10} - 2 \beta_{9} + 12 \beta_{8} + 30 \beta_{7} + 3 \beta_{6} + 18 \beta_{5} + 48 \beta_{4} - 40 \beta_{3} + 2 \beta_{2} - 135 \beta _1 - 8 ) / 48$$ (-23*b11 + 10*b10 - 2*b9 + 12*b8 + 30*b7 + 3*b6 + 18*b5 + 48*b4 - 40*b3 + 2*b2 - 135*b1 - 8) / 48 $$\nu^{9}$$ $$=$$ $$( -5\beta_{10} - \beta_{8} + 22\beta_{7} - 29\beta_{3} - 26\beta_{2} - 464 ) / 24$$ (-5*b10 - b8 + 22*b7 - 29*b3 - 26*b2 - 464) / 24 $$\nu^{10}$$ $$=$$ $$( - 43 \beta_{11} + 10 \beta_{10} - 82 \beta_{9} + 16 \beta_{8} - 30 \beta_{7} - 41 \beta_{6} - 14 \beta_{5} - 64 \beta_{4} - 28 \beta_{3} - 82 \beta_{2} + 237 \beta _1 - 152 ) / 48$$ (-43*b11 + 10*b10 - 82*b9 + 16*b8 - 30*b7 - 41*b6 - 14*b5 - 64*b4 - 28*b3 - 82*b2 + 237*b1 - 152) / 48 $$\nu^{11}$$ $$=$$ $$( 18 \beta_{11} - 33 \beta_{10} - 102 \beta_{9} - 5 \beta_{8} - 58 \beta_{7} + 122 \beta_{6} - 94 \beta_{5} - 20 \beta_{4} - 33 \beta_{3} + 102 \beta_{2} - 48 \beta _1 - 432 ) / 48$$ (18*b11 - 33*b10 - 102*b9 - 5*b8 - 58*b7 + 122*b6 - 94*b5 - 20*b4 - 33*b3 + 102*b2 - 48*b1 - 432) / 48

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 0.742163 + 1.20382i −1.41362 − 0.0408194i −1.07078 + 0.923815i 1.33544 − 0.465413i 0.578188 + 1.29062i 0.828615 + 1.14604i 0.828615 − 1.14604i 0.578188 − 1.29062i 1.33544 + 0.465413i −1.07078 − 0.923815i −1.41362 + 0.0408194i 0.742163 − 1.20382i
0 0 0 8.36964i 0 2.43910i 0 0 0
1567.2 0 0 0 8.36964i 0 2.43910i 0 0 0
1567.3 0 0 0 3.99718i 0 7.74742i 0 0 0
1567.4 0 0 0 3.99718i 0 7.74742i 0 0 0
1567.5 0 0 0 2.64040i 0 8.30833i 0 0 0
1567.6 0 0 0 2.64040i 0 8.30833i 0 0 0
1567.7 0 0 0 2.64040i 0 8.30833i 0 0 0
1567.8 0 0 0 2.64040i 0 8.30833i 0 0 0
1567.9 0 0 0 3.99718i 0 7.74742i 0 0 0
1567.10 0 0 0 3.99718i 0 7.74742i 0 0 0
1567.11 0 0 0 8.36964i 0 2.43910i 0 0 0
1567.12 0 0 0 8.36964i 0 2.43910i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1567.12 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
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.b.j yes 12
3.b odd 2 1 1728.3.b.i 12
4.b odd 2 1 inner 1728.3.b.j yes 12
8.b even 2 1 inner 1728.3.b.j yes 12
8.d odd 2 1 inner 1728.3.b.j yes 12
12.b even 2 1 1728.3.b.i 12
24.f even 2 1 1728.3.b.i 12
24.h odd 2 1 1728.3.b.i 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.b.i 12 3.b odd 2 1
1728.3.b.i 12 12.b even 2 1
1728.3.b.i 12 24.f even 2 1
1728.3.b.i 12 24.h odd 2 1
1728.3.b.j yes 12 1.a even 1 1 trivial
1728.3.b.j yes 12 4.b odd 2 1 inner
1728.3.b.j yes 12 8.b even 2 1 inner
1728.3.b.j yes 12 8.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{6} + 93T_{5}^{4} + 1719T_{5}^{2} + 7803$$ T5^6 + 93*T5^4 + 1719*T5^2 + 7803 $$T_{7}^{6} + 135T_{7}^{4} + 4911T_{7}^{2} + 24649$$ T7^6 + 135*T7^4 + 4911*T7^2 + 24649 $$T_{11}^{6} - 417T_{11}^{4} + 23211T_{11}^{2} - 121203$$ T11^6 - 417*T11^4 + 23211*T11^2 - 121203 $$T_{17}^{3} - 12T_{17}^{2} - 540T_{17} + 6912$$ T17^3 - 12*T17^2 - 540*T17 + 6912

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$(T^{6} + 93 T^{4} + 1719 T^{2} + \cdots + 7803)^{2}$$
$7$ $$(T^{6} + 135 T^{4} + 4911 T^{2} + \cdots + 24649)^{2}$$
$11$ $$(T^{6} - 417 T^{4} + 23211 T^{2} + \cdots - 121203)^{2}$$
$13$ $$(T^{6} + 936 T^{4} + 219024 T^{2} + \cdots + 442368)^{2}$$
$17$ $$(T^{3} - 12 T^{2} - 540 T + 6912)^{4}$$
$19$ $$(T^{6} - 1656 T^{4} + 671760 T^{2} + \cdots - 51121152)^{2}$$
$23$ $$(T^{6} + 2520 T^{4} + 1919376 T^{2} + \cdots + 394896384)^{2}$$
$29$ $$(T^{6} + 1512 T^{4} + 384912 T^{2} + \cdots + 20155392)^{2}$$
$31$ $$(T^{6} + 3447 T^{4} + 2857839 T^{2} + \cdots + 262018969)^{2}$$
$37$ $$(T^{6} + 5256 T^{4} + \cdots + 2674142208)^{2}$$
$41$ $$(T^{3} - 84 T^{2} - 972 T + 113184)^{4}$$
$43$ $$(T^{6} - 4320 T^{4} + 3172608 T^{2} + \cdots - 322486272)^{2}$$
$47$ $$(T^{6} + 7488 T^{4} + \cdots + 5780865024)^{2}$$
$53$ $$(T^{6} + 1101 T^{4} + 294615 T^{2} + \cdots + 8741547)^{2}$$
$59$ $$(T^{6} - 11232 T^{4} + \cdots - 5159780352)^{2}$$
$61$ $$(T^{2} + 768)^{6}$$
$67$ $$(T^{6} - 6336 T^{4} + 10036224 T^{2} + \cdots - 936050688)^{2}$$
$71$ $$(T^{6} + 22392 T^{4} + \cdots + 4087812096)^{2}$$
$73$ $$(T^{3} - 15 T^{2} - 5409 T - 120721)^{4}$$
$79$ $$(T^{6} + 8868 T^{4} + \cdots + 1846936576)^{2}$$
$83$ $$(T^{6} - 30801 T^{4} + \cdots - 11178011043)^{2}$$
$89$ $$(T^{3} - 312 T^{2} + 28944 T - 680832)^{4}$$
$97$ $$(T^{3} + 51 T^{2} - 22365 T - 1211279)^{4}$$