# Properties

 Label 729.2.c.c Level $729$ Weight $2$ Character orbit 729.c Analytic conductor $5.821$ 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] = [729,2,Mod(244,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("729.244");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.c (of order $$3$$, degree $$2$$, not 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: $$5.82109430735$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{36})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{6} + 1$$ x^12 - x^6 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{10} - \beta_{8} - \beta_{7}) q^{2} - \beta_{11} q^{4} + (\beta_{10} + \beta_{5} - \beta_{2}) q^{5} + ( - \beta_{11} + \beta_{9} + \beta_{6} - \beta_{3} + 2 \beta_1) q^{7} + ( - \beta_{8} - \beta_{7} + \beta_{5}) q^{8}+O(q^{10})$$ q + (b10 - b8 - b7) * q^2 - b11 * q^4 + (b10 + b5 - b2) * q^5 + (-b11 + b9 + b6 - b3 + 2*b1) * q^7 + (-b8 - b7 + b5) * q^8 $$q + (\beta_{10} - \beta_{8} - \beta_{7}) q^{2} - \beta_{11} q^{4} + (\beta_{10} + \beta_{5} - \beta_{2}) q^{5} + ( - \beta_{11} + \beta_{9} + \beta_{6} - \beta_{3} + 2 \beta_1) q^{7} + ( - \beta_{8} - \beta_{7} + \beta_{5}) q^{8} + ( - 2 \beta_{9} - 3 \beta_{6} - 2) q^{10} + (\beta_{4} - 2 \beta_{2}) q^{11} + ( - \beta_{11} + 3 \beta_{6} - 3 \beta_{3} - 2 \beta_1 + 2) q^{13} + (3 \beta_{10} - \beta_{7} + \beta_{4}) q^{14} + ( - 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{6} - \beta_{3} + 2 \beta_1) q^{16} + ( - 2 \beta_{8} + 2 \beta_{7} + \beta_{5}) q^{17} + ( - 2 \beta_{9} - \beta_{6} - 4) q^{19} + ( - 2 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} + \beta_{4} + \beta_{2}) q^{20} + ( - 2 \beta_{11} - \beta_{6} + \beta_{3} - \beta_1 + 1) q^{22} + ( - 3 \beta_{10} + 5 \beta_{7} - 5 \beta_{4}) q^{23} + (3 \beta_{11} - 3 \beta_{9} - 3 \beta_{6} - 2 \beta_{3}) q^{25} + ( - \beta_{8} - 4 \beta_{7} - 2 \beta_{5}) q^{26} + ( - \beta_{9} - 2 \beta_{6} - 1) q^{28} + (2 \beta_{10} - 2 \beta_{8} - 2 \beta_{7} - \beta_{4} + \beta_{2}) q^{29} + ( - 2 \beta_{11} + 2 \beta_{6} - 2 \beta_{3} - 5 \beta_1 + 5) q^{31} + (2 \beta_{10} - \beta_{7} + 3 \beta_{5} + \beta_{4} - 3 \beta_{2}) q^{32} + ( - \beta_{11} + \beta_{9} + \beta_{6} - 5 \beta_{3}) q^{34} + (2 \beta_{8} + 3 \beta_{7} + 2 \beta_{5}) q^{35} + (\beta_{9} + 2 \beta_{6} - 1) q^{37} + ( - 6 \beta_{10} + 6 \beta_{8} + 6 \beta_{7} - \beta_{4} + \beta_{2}) q^{38} + ( - \beta_{11} - \beta_1 + 1) q^{40} + (3 \beta_{10} - 4 \beta_{7} + \beta_{5} + 4 \beta_{4} - \beta_{2}) q^{41} + ( - 2 \beta_{3} + 2 \beta_1) q^{43} + (\beta_{8} + 4 \beta_{7} - \beta_{5}) q^{44} + (3 \beta_{9} - 2 \beta_{6} + 1) q^{46} + (\beta_{10} - \beta_{8} - \beta_{7} + \beta_{2}) q^{47} + ( - 3 \beta_{11} + 4 \beta_{6} - 4 \beta_{3} - \beta_1 + 1) q^{49} + ( - 3 \beta_{10} - 2 \beta_{7} - 5 \beta_{5} + 2 \beta_{4} + 5 \beta_{2}) q^{50} + ( - 5 \beta_{11} + 5 \beta_{9} + 5 \beta_{6} - \beta_{3} + \beta_1) q^{52} + (2 \beta_{8} - 2 \beta_{7} - 2 \beta_{5}) q^{53} + ( - 2 \beta_{6} - 5) q^{55} + (4 \beta_{10} - 4 \beta_{8} - 4 \beta_{7} + 3 \beta_{4} + 2 \beta_{2}) q^{56} + ( - \beta_{11} + 5 \beta_1 - 5) q^{58} + ( - 5 \beta_{10} - 2 \beta_{5} + 2 \beta_{2}) q^{59} + (4 \beta_{11} - 4 \beta_{9} - 4 \beta_{6} + 6 \beta_{3} + 2 \beta_1) q^{61} + ( - 3 \beta_{8} - 5 \beta_{7}) q^{62} + ( - \beta_{9} - 5 \beta_{6} + 1) q^{64} + (6 \beta_{10} - 6 \beta_{8} - 6 \beta_{7} + \beta_{4} - 4 \beta_{2}) q^{65} + ( - 3 \beta_{11} - 5 \beta_{6} + 5 \beta_{3} + \beta_1 - 1) q^{67} + ( - 3 \beta_{10} + 3 \beta_{7} - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{2}) q^{68} + (4 \beta_{11} - 4 \beta_{9} - 4 \beta_{6} - 3 \beta_{3} - 5 \beta_1) q^{70} + (2 \beta_{8} - 2 \beta_{7}) q^{71} + ( - 6 \beta_{9} - 3 \beta_{6} + 2) q^{73} + ( - \beta_{4} - 2 \beta_{2}) q^{74} + (3 \beta_{11} + 2 \beta_{6} - 2 \beta_{3} - 3 \beta_1 + 3) q^{76} + ( - 2 \beta_{10} + 3 \beta_{7} + 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{2}) q^{77} + ( - 5 \beta_{11} + 5 \beta_{9} + 5 \beta_{6} - 3 \beta_{3} + 8 \beta_1) q^{79} + (4 \beta_{8} + 6 \beta_{7} + 3 \beta_{5}) q^{80} + ( - 4 \beta_{9} - \beta_{6} - 2) q^{82} + ( - 3 \beta_{10} + 3 \beta_{8} + 3 \beta_{7} - 8 \beta_{4} - 4 \beta_{2}) q^{83} + (5 \beta_{11} - \beta_{6} + \beta_{3} - 3 \beta_1 + 3) q^{85} + (2 \beta_{10} - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{86} + ( - 4 \beta_{11} + 4 \beta_{9} + 4 \beta_{6} - 7 \beta_1) q^{88} + (6 \beta_{8} + 3 \beta_{7} - 5 \beta_{5}) q^{89} + (\beta_{9} + 7 \beta_{6}) q^{91} + ( - 2 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} - 5 \beta_{4} + 2 \beta_{2}) q^{92} + (\beta_{6} - \beta_{3} + 2 \beta_1 - 2) q^{94} + ( - 6 \beta_{10} - 2 \beta_{7} - 5 \beta_{5} + 2 \beta_{4} + 5 \beta_{2}) q^{95} + (3 \beta_{11} - 3 \beta_{9} - 3 \beta_{6} + 7 \beta_{3} + 2 \beta_1) q^{97} + (2 \beta_{8} - 2 \beta_{7} - \beta_{5}) q^{98}+O(q^{100})$$ q + (b10 - b8 - b7) * q^2 - b11 * q^4 + (b10 + b5 - b2) * q^5 + (-b11 + b9 + b6 - b3 + 2*b1) * q^7 + (-b8 - b7 + b5) * q^8 + (-2*b9 - 3*b6 - 2) * q^10 + (b4 - 2*b2) * q^11 + (-b11 + 3*b6 - 3*b3 - 2*b1 + 2) * q^13 + (3*b10 - b7 + b4) * q^14 + (-2*b11 + 2*b9 + 2*b6 - b3 + 2*b1) * q^16 + (-2*b8 + 2*b7 + b5) * q^17 + (-2*b9 - b6 - 4) * q^19 + (-2*b10 + 2*b8 + 2*b7 + b4 + b2) * q^20 + (-2*b11 - b6 + b3 - b1 + 1) * q^22 + (-3*b10 + 5*b7 - 5*b4) * q^23 + (3*b11 - 3*b9 - 3*b6 - 2*b3) * q^25 + (-b8 - 4*b7 - 2*b5) * q^26 + (-b9 - 2*b6 - 1) * q^28 + (2*b10 - 2*b8 - 2*b7 - b4 + b2) * q^29 + (-2*b11 + 2*b6 - 2*b3 - 5*b1 + 5) * q^31 + (2*b10 - b7 + 3*b5 + b4 - 3*b2) * q^32 + (-b11 + b9 + b6 - 5*b3) * q^34 + (2*b8 + 3*b7 + 2*b5) * q^35 + (b9 + 2*b6 - 1) * q^37 + (-6*b10 + 6*b8 + 6*b7 - b4 + b2) * q^38 + (-b11 - b1 + 1) * q^40 + (3*b10 - 4*b7 + b5 + 4*b4 - b2) * q^41 + (-2*b3 + 2*b1) * q^43 + (b8 + 4*b7 - b5) * q^44 + (3*b9 - 2*b6 + 1) * q^46 + (b10 - b8 - b7 + b2) * q^47 + (-3*b11 + 4*b6 - 4*b3 - b1 + 1) * q^49 + (-3*b10 - 2*b7 - 5*b5 + 2*b4 + 5*b2) * q^50 + (-5*b11 + 5*b9 + 5*b6 - b3 + b1) * q^52 + (2*b8 - 2*b7 - 2*b5) * q^53 + (-2*b6 - 5) * q^55 + (4*b10 - 4*b8 - 4*b7 + 3*b4 + 2*b2) * q^56 + (-b11 + 5*b1 - 5) * q^58 + (-5*b10 - 2*b5 + 2*b2) * q^59 + (4*b11 - 4*b9 - 4*b6 + 6*b3 + 2*b1) * q^61 + (-3*b8 - 5*b7) * q^62 + (-b9 - 5*b6 + 1) * q^64 + (6*b10 - 6*b8 - 6*b7 + b4 - 4*b2) * q^65 + (-3*b11 - 5*b6 + 5*b3 + b1 - 1) * q^67 + (-3*b10 + 3*b7 - 2*b5 - 3*b4 + 2*b2) * q^68 + (4*b11 - 4*b9 - 4*b6 - 3*b3 - 5*b1) * q^70 + (2*b8 - 2*b7) * q^71 + (-6*b9 - 3*b6 + 2) * q^73 + (-b4 - 2*b2) * q^74 + (3*b11 + 2*b6 - 2*b3 - 3*b1 + 3) * q^76 + (-2*b10 + 3*b7 + 3*b5 - 3*b4 - 3*b2) * q^77 + (-5*b11 + 5*b9 + 5*b6 - 3*b3 + 8*b1) * q^79 + (4*b8 + 6*b7 + 3*b5) * q^80 + (-4*b9 - b6 - 2) * q^82 + (-3*b10 + 3*b8 + 3*b7 - 8*b4 - 4*b2) * q^83 + (5*b11 - b6 + b3 - 3*b1 + 3) * q^85 + (2*b10 - 2*b7 - 2*b5 + 2*b4 + 2*b2) * q^86 + (-4*b11 + 4*b9 + 4*b6 - 7*b1) * q^88 + (6*b8 + 3*b7 - 5*b5) * q^89 + (b9 + 7*b6) * q^91 + (-2*b10 + 2*b8 + 2*b7 - 5*b4 + 2*b2) * q^92 + (b6 - b3 + 2*b1 - 2) * q^94 + (-6*b10 - 2*b7 - 5*b5 + 2*b4 + 5*b2) * q^95 + (3*b11 - 3*b9 - 3*b6 + 7*b3 + 2*b1) * q^97 + (2*b8 - 2*b7 - b5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 12 q^{7}+O(q^{10})$$ 12 * q + 12 * q^7 $$12 q + 12 q^{7} - 24 q^{10} + 12 q^{13} + 12 q^{16} - 48 q^{19} + 6 q^{22} - 12 q^{28} + 30 q^{31} - 12 q^{37} + 6 q^{40} + 12 q^{43} + 12 q^{46} + 6 q^{49} + 6 q^{52} - 60 q^{55} - 30 q^{58} + 12 q^{61} + 12 q^{64} - 6 q^{67} - 30 q^{70} + 24 q^{73} + 18 q^{76} + 48 q^{79} - 24 q^{82} + 18 q^{85} - 42 q^{88} - 12 q^{94} + 12 q^{97}+O(q^{100})$$ 12 * q + 12 * q^7 - 24 * q^10 + 12 * q^13 + 12 * q^16 - 48 * q^19 + 6 * q^22 - 12 * q^28 + 30 * q^31 - 12 * q^37 + 6 * q^40 + 12 * q^43 + 12 * q^46 + 6 * q^49 + 6 * q^52 - 60 * q^55 - 30 * q^58 + 12 * q^61 + 12 * q^64 - 6 * q^67 - 30 * q^70 + 24 * q^73 + 18 * q^76 + 48 * q^79 - 24 * q^82 + 18 * q^85 - 42 * q^88 - 12 * q^94 + 12 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{36}^{6}$$ v^6 $$\beta_{2}$$ $$=$$ $$\zeta_{36}^{9} + \zeta_{36}^{3}$$ v^9 + v^3 $$\beta_{3}$$ $$=$$ $$\zeta_{36}^{10} + \zeta_{36}^{2}$$ v^10 + v^2 $$\beta_{4}$$ $$=$$ $$\zeta_{36}^{11} + \zeta_{36}$$ v^11 + v $$\beta_{5}$$ $$=$$ $$-\zeta_{36}^{9} + 2\zeta_{36}^{3}$$ -v^9 + 2*v^3 $$\beta_{6}$$ $$=$$ $$-\zeta_{36}^{8} + \zeta_{36}^{4} + \zeta_{36}^{2}$$ -v^8 + v^4 + v^2 $$\beta_{7}$$ $$=$$ $$-\zeta_{36}^{7} + \zeta_{36}^{5} + \zeta_{36}$$ -v^7 + v^5 + v $$\beta_{8}$$ $$=$$ $$-\zeta_{36}^{11} + \zeta_{36}^{7}$$ -v^11 + v^7 $$\beta_{9}$$ $$=$$ $$-\zeta_{36}^{10} + \zeta_{36}^{8}$$ -v^10 + v^8 $$\beta_{10}$$ $$=$$ $$-\zeta_{36}^{11} - \zeta_{36}^{7} + \zeta_{36}$$ -v^11 - v^7 + v $$\beta_{11}$$ $$=$$ $$-\zeta_{36}^{10} - \zeta_{36}^{8} + \zeta_{36}^{2}$$ -v^10 - v^8 + v^2
 $$\zeta_{36}$$ $$=$$ $$( \beta_{10} + \beta_{8} + 2\beta_{4} ) / 3$$ (b10 + b8 + 2*b4) / 3 $$\zeta_{36}^{2}$$ $$=$$ $$( \beta_{11} + \beta_{9} + 2\beta_{3} ) / 3$$ (b11 + b9 + 2*b3) / 3 $$\zeta_{36}^{3}$$ $$=$$ $$( \beta_{5} + \beta_{2} ) / 3$$ (b5 + b2) / 3 $$\zeta_{36}^{4}$$ $$=$$ $$( -2\beta_{11} + \beta_{9} + 3\beta_{6} - \beta_{3} ) / 3$$ (-2*b11 + b9 + 3*b6 - b3) / 3 $$\zeta_{36}^{5}$$ $$=$$ $$( -2\beta_{10} + \beta_{8} + 3\beta_{7} - \beta_{4} ) / 3$$ (-2*b10 + b8 + 3*b7 - b4) / 3 $$\zeta_{36}^{6}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{36}^{7}$$ $$=$$ $$( -\beta_{10} + 2\beta_{8} + \beta_{4} ) / 3$$ (-b10 + 2*b8 + b4) / 3 $$\zeta_{36}^{8}$$ $$=$$ $$( -\beta_{11} + 2\beta_{9} + \beta_{3} ) / 3$$ (-b11 + 2*b9 + b3) / 3 $$\zeta_{36}^{9}$$ $$=$$ $$( -\beta_{5} + 2\beta_{2} ) / 3$$ (-b5 + 2*b2) / 3 $$\zeta_{36}^{10}$$ $$=$$ $$( -\beta_{11} - \beta_{9} + \beta_{3} ) / 3$$ (-b11 - b9 + b3) / 3 $$\zeta_{36}^{11}$$ $$=$$ $$( -\beta_{10} - \beta_{8} + \beta_{4} ) / 3$$ (-b10 - b8 + b4) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
244.1
 0.984808 + 0.173648i 0.642788 − 0.766044i 0.342020 + 0.939693i −0.342020 − 0.939693i −0.642788 + 0.766044i −0.984808 − 0.173648i 0.984808 − 0.173648i 0.642788 + 0.766044i 0.342020 − 0.939693i −0.342020 + 0.939693i −0.642788 − 0.766044i −0.984808 + 0.173648i
−0.984808 1.70574i 0 −0.939693 + 1.62760i 1.85083 3.20574i 0 1.17365 + 2.03282i −0.237565 0 −7.29086
244.2 −0.642788 1.11334i 0 0.173648 0.300767i −0.223238 + 0.386659i 0 1.76604 + 3.05888i −3.01763 0 0.573978
244.3 −0.342020 0.592396i 0 0.766044 1.32683i −0.524005 + 0.907604i 0 0.0603074 + 0.104455i −2.41609 0 0.716881
244.4 0.342020 + 0.592396i 0 0.766044 1.32683i 0.524005 0.907604i 0 0.0603074 + 0.104455i 2.41609 0 0.716881
244.5 0.642788 + 1.11334i 0 0.173648 0.300767i 0.223238 0.386659i 0 1.76604 + 3.05888i 3.01763 0 0.573978
244.6 0.984808 + 1.70574i 0 −0.939693 + 1.62760i −1.85083 + 3.20574i 0 1.17365 + 2.03282i 0.237565 0 −7.29086
487.1 −0.984808 + 1.70574i 0 −0.939693 1.62760i 1.85083 + 3.20574i 0 1.17365 2.03282i −0.237565 0 −7.29086
487.2 −0.642788 + 1.11334i 0 0.173648 + 0.300767i −0.223238 0.386659i 0 1.76604 3.05888i −3.01763 0 0.573978
487.3 −0.342020 + 0.592396i 0 0.766044 + 1.32683i −0.524005 0.907604i 0 0.0603074 0.104455i −2.41609 0 0.716881
487.4 0.342020 0.592396i 0 0.766044 + 1.32683i 0.524005 + 0.907604i 0 0.0603074 0.104455i 2.41609 0 0.716881
487.5 0.642788 1.11334i 0 0.173648 + 0.300767i 0.223238 + 0.386659i 0 1.76604 3.05888i 3.01763 0 0.573978
487.6 0.984808 1.70574i 0 −0.939693 1.62760i −1.85083 3.20574i 0 1.17365 2.03282i 0.237565 0 −7.29086
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 487.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.c.c 12
3.b odd 2 1 inner 729.2.c.c 12
9.c even 3 1 729.2.a.c 6
9.c even 3 1 inner 729.2.c.c 12
9.d odd 6 1 729.2.a.c 6
9.d odd 6 1 inner 729.2.c.c 12
27.e even 9 2 729.2.e.m 12
27.e even 9 2 729.2.e.q 12
27.e even 9 2 729.2.e.r 12
27.f odd 18 2 729.2.e.m 12
27.f odd 18 2 729.2.e.q 12
27.f odd 18 2 729.2.e.r 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.2.a.c 6 9.c even 3 1
729.2.a.c 6 9.d odd 6 1
729.2.c.c 12 1.a even 1 1 trivial
729.2.c.c 12 3.b odd 2 1 inner
729.2.c.c 12 9.c even 3 1 inner
729.2.c.c 12 9.d odd 6 1 inner
729.2.e.m 12 27.e even 9 2
729.2.e.m 12 27.f odd 18 2
729.2.e.q 12 27.e even 9 2
729.2.e.q 12 27.f odd 18 2
729.2.e.r 12 27.e even 9 2
729.2.e.r 12 27.f odd 18 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 6T_{2}^{10} + 27T_{2}^{8} + 48T_{2}^{6} + 63T_{2}^{4} + 27T_{2}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(729, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 6 T^{10} + 27 T^{8} + 48 T^{6} + \cdots + 9$$
$3$ $$T^{12}$$
$5$ $$T^{12} + 15 T^{10} + 207 T^{8} + 264 T^{6} + \cdots + 9$$
$7$ $$(T^{6} - 6 T^{5} + 27 T^{4} - 52 T^{3} + \cdots + 1)^{2}$$
$11$ $$T^{12} + 42 T^{10} + 1359 T^{8} + \cdots + 1172889$$
$13$ $$(T^{6} - 6 T^{5} + 45 T^{4} - 88 T^{3} + \cdots + 5041)^{2}$$
$17$ $$(T^{6} - 81 T^{4} + 1755 T^{2} + \cdots - 9747)^{2}$$
$19$ $$(T^{3} + 12 T^{2} + 39 T + 19)^{4}$$
$23$ $$T^{12} + 114 T^{10} + 9747 T^{8} + \cdots + 751689$$
$29$ $$T^{12} + 51 T^{10} + 1791 T^{8} + \cdots + 16867449$$
$31$ $$(T^{6} - 15 T^{5} + 162 T^{4} - 799 T^{3} + \cdots + 5329)^{2}$$
$37$ $$(T^{3} + 3 T^{2} - 6 T - 17)^{4}$$
$41$ $$T^{12} + 87 T^{10} + 5715 T^{8} + \cdots + 71014329$$
$43$ $$(T^{6} - 6 T^{5} + 36 T^{4} - 16 T^{3} + \cdots + 64)^{2}$$
$47$ $$T^{12} + 15 T^{10} + 171 T^{8} + 804 T^{6} + \cdots + 9$$
$53$ $$(T^{6} - 108 T^{4} + 2592 T^{2} + \cdots - 15552)^{2}$$
$59$ $$T^{12} + 186 T^{10} + 33039 T^{8} + \cdots + 9$$
$61$ $$(T^{6} - 6 T^{5} + 108 T^{4} - 160 T^{3} + \cdots + 87616)^{2}$$
$67$ $$(T^{6} + 3 T^{5} + 153 T^{4} - 934 T^{3} + \cdots + 63001)^{2}$$
$71$ $$(T^{6} - 72 T^{4} + 1296 T^{2} + \cdots - 1728)^{2}$$
$73$ $$(T^{3} - 6 T^{2} - 69 T - 89)^{4}$$
$79$ $$(T^{6} - 24 T^{5} + 441 T^{4} + \cdots + 11449)^{2}$$
$83$ $$T^{12} + 438 T^{10} + \cdots + 2405238079689$$
$89$ $$(T^{6} - 387 T^{4} + 16146 T^{2} + \cdots - 7803)^{2}$$
$97$ $$(T^{6} - 6 T^{5} + 135 T^{4} + \cdots + 418609)^{2}$$