# Properties

 Label 819.2.j.e Level $819$ Weight $2$ Character orbit 819.j Analytic conductor $6.540$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(235,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("819.235");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.j (of order $$3$$, degree $$2$$, 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: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) 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 primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} + \zeta_{18}) q^{2} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}) q^{4} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{5} + ( - \zeta_{18}^{4} - 2 \zeta_{18}) q^{7} + (\zeta_{18}^{5} - \zeta_{18}^{4} - 1) q^{8}+O(q^{10})$$ q + (z^5 - z^4 - z^2 + z) * q^2 + (-z^5 - z^4 + z) * q^4 + (z^5 - z^4 + z^3 - z^2 + z) * q^5 + (-z^4 - 2*z) * q^7 + (z^5 - z^4 - 1) * q^8 $$q + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} + \zeta_{18}) q^{2} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}) q^{4} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{5} + ( - \zeta_{18}^{4} - 2 \zeta_{18}) q^{7} + (\zeta_{18}^{5} - \zeta_{18}^{4} - 1) q^{8} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{3} - \zeta_{18}^{2} + 2 \zeta_{18} - 2) q^{10} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{11} + q^{13} + (2 \zeta_{18}^{5} + \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 2) q^{14} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - \zeta_{18}) q^{16} + (\zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{3} - \zeta_{18} + 3) q^{17} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{19} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 1) q^{20} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 1) q^{22} + ( - 4 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} - 4 \zeta_{18}) q^{23} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18} + 2) q^{25} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} + \zeta_{18}) q^{26} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 3) q^{28} + ( - \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 4 \zeta_{18}^{2} - 4 \zeta_{18} - 2) q^{29} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18} - 4) q^{31} + (\zeta_{18}^{5} + \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 2 \zeta_{18} - 3) q^{32} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{4} + 1) q^{34} + (2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18} + 2) q^{35} + (\zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2}) q^{37} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + \zeta_{18}^{2} - 3 \zeta_{18} + 4) q^{38} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2}) q^{40} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{4}) q^{41} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{2} + 3 \zeta_{18} - 2) q^{43} - \zeta_{18}^{3} q^{44} + (4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + \zeta_{18} + 3) q^{46} + ( - 4 \zeta_{18}^{5} + 2 \zeta_{18}^{3} - 4 \zeta_{18}) q^{47} + (5 \zeta_{18}^{5} + 3 \zeta_{18}^{2}) q^{49} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18} - 5) q^{50} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}) q^{52} + (\zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} - 1) q^{53} + ( - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{55} + (3 \zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \zeta_{18}^{2} + 2 \zeta_{18} + 3) q^{56} + (2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 6 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 2 \zeta_{18}) q^{58} + ( - 8 \zeta_{18}^{5} - 8 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + 9 \zeta_{18} - 1) q^{59} + ( - 8 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - 8 \zeta_{18}) q^{61} + ( - 5 \zeta_{18}^{5} + 6 \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18}) q^{62} + ( - \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 3 \zeta_{18}^{2} - 3 \zeta_{18} - 1) q^{64} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{65} + (5 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 3) q^{67} + (\zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18}) q^{68} + (\zeta_{18}^{5} - 4 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 6 \zeta_{18} - 4) q^{70} + ( - 5 \zeta_{18}^{5} - \zeta_{18}^{4} + 6 \zeta_{18}^{2} + 6 \zeta_{18} - 7) q^{71} + (6 \zeta_{18}^{5} + 6 \zeta_{18}^{4} + \zeta_{18}^{2} - 7 \zeta_{18}) q^{73} + ( - \zeta_{18}^{3} + 1) q^{74} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 2) q^{76} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} + 3 \zeta_{18} - 2) q^{77} + ( - \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 8 \zeta_{18}^{3} - 5 \zeta_{18}^{2} - \zeta_{18}) q^{79} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 4 \zeta_{18}^{2} + 6 \zeta_{18} - 3) q^{80} + (3 \zeta_{18}^{4} - 6 \zeta_{18}^{3} + 3 \zeta_{18}^{2}) q^{82} + (\zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 4) q^{83} + (5 \zeta_{18}^{5} - 4 \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 4) q^{85} + ( - 5 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 5 \zeta_{18}) q^{86} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18} + 2) q^{88} + ( - \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 5 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - \zeta_{18}) q^{89} + ( - \zeta_{18}^{4} - 2 \zeta_{18}) q^{91} + (4 \zeta_{18}^{5} + \zeta_{18}^{4} - 5 \zeta_{18}^{2} - 5 \zeta_{18} - 6) q^{92} + (4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 6 \zeta_{18}^{2} + 2 \zeta_{18} + 4) q^{94} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 5 \zeta_{18} + 5) q^{95} + ( - 5 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} + 7) q^{97} + ( - 5 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 3 \zeta_{18} + 3) q^{98} +O(q^{100})$$ q + (z^5 - z^4 - z^2 + z) * q^2 + (-z^5 - z^4 + z) * q^4 + (z^5 - z^4 + z^3 - z^2 + z) * q^5 + (-z^4 - 2*z) * q^7 + (z^5 - z^4 - 1) * q^8 + (-z^5 - z^4 + 2*z^3 - z^2 + 2*z - 2) * q^10 + (-z^5 - z^4 + z^3 + z^2 - 1) * q^11 + q^13 + (2*z^5 + z^3 - 3*z^2 + 2) * q^14 + (-z^5 - 2*z^4 + 2*z^3 - 2*z^2 - z) * q^16 + (z^5 + z^4 - 3*z^3 - z + 3) * q^17 + (-2*z^5 + 2*z^4 - z^3 + 2*z^2 - 2*z) * q^19 + (-2*z^5 + z^4 + z^2 + z - 1) * q^20 + (-z^5 + 2*z^4 - z^2 - z + 1) * q^22 + (-4*z^5 - z^4 - z^2 - 4*z) * q^23 + (-z^5 - z^4 - 2*z^3 - 2*z^2 + 3*z + 2) * q^25 + (z^5 - z^4 - z^2 + z) * q^26 + (2*z^5 + 2*z^3 - 3*z^2 - 3) * q^28 + (-z^5 + 5*z^4 - 4*z^2 - 4*z - 2) * q^29 + (-2*z^5 - 2*z^4 + 4*z^3 + z^2 + z - 4) * q^31 + (z^5 + z^4 + 3*z^3 - 3*z^2 + 2*z - 3) * q^32 + (4*z^5 - 4*z^4 + 1) * q^34 + (2*z^5 - 3*z^4 + z^3 - 3*z^2 + z + 2) * q^35 + (z^4 + z^3 + z^2) * q^37 + (2*z^5 + 2*z^4 - 4*z^3 + z^2 - 3*z + 4) * q^38 + (-z^4 + z^3 - z^2) * q^40 + (-3*z^5 + 3*z^4) * q^41 + (-z^5 - 2*z^4 + 3*z^2 + 3*z - 2) * q^43 - z^3 * q^44 + (4*z^5 + 4*z^4 - 3*z^3 - 5*z^2 + z + 3) * q^46 + (-4*z^5 + 2*z^3 - 4*z) * q^47 + (5*z^5 + 3*z^2) * q^49 + (-z^5 - z^4 + 2*z^2 + 2*z - 5) * q^50 + (-z^5 - z^4 + z) * q^52 + (z^5 + z^4 + z^3 - z^2 - 1) * q^53 + (-z^5 + z^4) * q^55 + (3*z^5 + z^4 - 2*z^3 - z^2 + 2*z + 3) * q^56 + (2*z^5 + 3*z^4 - 6*z^3 + 3*z^2 + 2*z) * q^58 + (-8*z^5 - 8*z^4 + z^3 - z^2 + 9*z - 1) * q^59 + (-8*z^5 + 5*z^4 - 5*z^3 + 5*z^2 - 8*z) * q^61 + (-5*z^5 + 6*z^4 - z^2 - z) * q^62 + (-z^5 + 4*z^4 - 3*z^2 - 3*z - 1) * q^64 + (z^5 - z^4 + z^3 - z^2 + z) * q^65 + (5*z^5 + 5*z^4 - 3*z^3 - 5*z^2 + 3) * q^67 + (z^5 - 3*z^4 + 2*z^3 - 3*z^2 + z) * q^68 + (z^5 - 4*z^4 + 5*z^3 - 5*z^2 + 6*z - 4) * q^70 + (-5*z^5 - z^4 + 6*z^2 + 6*z - 7) * q^71 + (6*z^5 + 6*z^4 + z^2 - 7*z) * q^73 + (-z^3 + 1) * q^74 + (3*z^5 - 2*z^4 - z^2 - z + 2) * q^76 + (3*z^5 - 2*z^4 - z^3 - z^2 + 3*z - 2) * q^77 + (-z^5 - 5*z^4 + 8*z^3 - 5*z^2 - z) * q^79 + (-2*z^5 - 2*z^4 + 3*z^3 - 4*z^2 + 6*z - 3) * q^80 + (3*z^4 - 6*z^3 + 3*z^2) * q^82 + (z^4 - z^2 - z + 4) * q^83 + (5*z^5 - 4*z^4 - z^2 - z + 4) * q^85 + (-5*z^5 + 3*z^4 + z^3 + 3*z^2 - 5*z) * q^86 + (2*z^5 + 2*z^4 - 2*z^3 - 3*z^2 + z + 2) * q^88 + (-z^5 + 4*z^4 + 5*z^3 + 4*z^2 - z) * q^89 + (-z^4 - 2*z) * q^91 + (4*z^5 + z^4 - 5*z^2 - 5*z - 6) * q^92 + (4*z^5 + 4*z^4 - 4*z^3 - 6*z^2 + 2*z + 4) * q^94 + (2*z^5 + 2*z^4 - 5*z^3 + 3*z^2 - 5*z + 5) * q^95 + (-5*z^5 + 4*z^4 + z^2 + z + 7) * q^97 + (-5*z^4 + 5*z^3 - 3*z + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{5} - 6 q^{8}+O(q^{10})$$ 6 * q + 3 * q^5 - 6 * q^8 $$6 q + 3 q^{5} - 6 q^{8} - 6 q^{10} - 3 q^{11} + 6 q^{13} + 15 q^{14} + 6 q^{16} + 9 q^{17} - 3 q^{19} - 6 q^{20} + 6 q^{22} + 6 q^{25} - 12 q^{28} - 12 q^{29} - 12 q^{31} - 9 q^{32} + 6 q^{34} + 15 q^{35} + 3 q^{37} + 12 q^{38} + 3 q^{40} - 12 q^{43} - 3 q^{44} + 9 q^{46} + 6 q^{47} - 30 q^{50} - 3 q^{53} + 12 q^{56} - 18 q^{58} - 3 q^{59} - 15 q^{61} - 6 q^{64} + 3 q^{65} + 9 q^{67} + 6 q^{68} - 9 q^{70} - 42 q^{71} + 3 q^{74} + 12 q^{76} - 15 q^{77} + 24 q^{79} - 9 q^{80} - 18 q^{82} + 24 q^{83} + 24 q^{85} + 3 q^{86} + 6 q^{88} + 15 q^{89} - 36 q^{92} + 12 q^{94} + 15 q^{95} + 42 q^{97} + 33 q^{98}+O(q^{100})$$ 6 * q + 3 * q^5 - 6 * q^8 - 6 * q^10 - 3 * q^11 + 6 * q^13 + 15 * q^14 + 6 * q^16 + 9 * q^17 - 3 * q^19 - 6 * q^20 + 6 * q^22 + 6 * q^25 - 12 * q^28 - 12 * q^29 - 12 * q^31 - 9 * q^32 + 6 * q^34 + 15 * q^35 + 3 * q^37 + 12 * q^38 + 3 * q^40 - 12 * q^43 - 3 * q^44 + 9 * q^46 + 6 * q^47 - 30 * q^50 - 3 * q^53 + 12 * q^56 - 18 * q^58 - 3 * q^59 - 15 * q^61 - 6 * q^64 + 3 * q^65 + 9 * q^67 + 6 * q^68 - 9 * q^70 - 42 * q^71 + 3 * q^74 + 12 * q^76 - 15 * q^77 + 24 * q^79 - 9 * q^80 - 18 * q^82 + 24 * q^83 + 24 * q^85 + 3 * q^86 + 6 * q^88 + 15 * q^89 - 36 * q^92 + 12 * q^94 + 15 * q^95 + 42 * q^97 + 33 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{18}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
235.1
 −0.173648 − 0.984808i 0.939693 + 0.342020i −0.766044 + 0.642788i −0.173648 + 0.984808i 0.939693 − 0.342020i −0.766044 − 0.642788i
−0.766044 1.32683i 0 −0.173648 + 0.300767i −0.266044 0.460802i 0 −0.418748 + 2.61240i −2.53209 0 −0.407604 + 0.705990i
235.2 −0.173648 0.300767i 0 0.939693 1.62760i 0.326352 + 0.565258i 0 −2.05303 1.66885i −1.34730 0 0.113341 0.196312i
235.3 0.939693 + 1.62760i 0 −0.766044 + 1.32683i 1.43969 + 2.49362i 0 2.47178 0.943555i 0.879385 0 −2.70574 + 4.68647i
352.1 −0.766044 + 1.32683i 0 −0.173648 0.300767i −0.266044 + 0.460802i 0 −0.418748 2.61240i −2.53209 0 −0.407604 0.705990i
352.2 −0.173648 + 0.300767i 0 0.939693 + 1.62760i 0.326352 0.565258i 0 −2.05303 + 1.66885i −1.34730 0 0.113341 + 0.196312i
352.3 0.939693 1.62760i 0 −0.766044 1.32683i 1.43969 2.49362i 0 2.47178 + 0.943555i 0.879385 0 −2.70574 4.68647i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 235.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.j.e 6
3.b odd 2 1 273.2.i.b 6
7.c even 3 1 inner 819.2.j.e 6
7.c even 3 1 5733.2.a.y 3
7.d odd 6 1 5733.2.a.z 3
21.g even 6 1 1911.2.a.p 3
21.h odd 6 1 273.2.i.b 6
21.h odd 6 1 1911.2.a.o 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.i.b 6 3.b odd 2 1
273.2.i.b 6 21.h odd 6 1
819.2.j.e 6 1.a even 1 1 trivial
819.2.j.e 6 7.c even 3 1 inner
1911.2.a.o 3 21.h odd 6 1
1911.2.a.p 3 21.g even 6 1
5733.2.a.y 3 7.c even 3 1
5733.2.a.z 3 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 3T_{2}^{4} + 2T_{2}^{3} + 9T_{2}^{2} + 3T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$3$ $$T^{6}$$
$5$ $$T^{6} - 3 T^{5} + 9 T^{4} - 2 T^{3} + \cdots + 1$$
$7$ $$T^{6} - 17T^{3} + 343$$
$11$ $$T^{6} + 3 T^{5} + 9 T^{4} + 2 T^{3} + \cdots + 1$$
$13$ $$(T - 1)^{6}$$
$17$ $$T^{6} - 9 T^{5} + 57 T^{4} - 182 T^{3} + \cdots + 289$$
$19$ $$T^{6} + 3 T^{5} + 18 T^{4} - 21 T^{3} + \cdots + 9$$
$23$ $$T^{6} + 63 T^{4} - 18 T^{3} + 3969 T^{2} + \cdots + 81$$
$29$ $$(T^{3} + 6 T^{2} - 51 T - 289)^{2}$$
$31$ $$T^{6} + 12 T^{5} + 105 T^{4} + \cdots + 361$$
$37$ $$T^{6} - 3 T^{5} + 9 T^{4} - 2 T^{3} + \cdots + 1$$
$41$ $$(T^{3} - 27 T + 27)^{2}$$
$43$ $$(T^{3} + 6 T^{2} - 9 T - 51)^{2}$$
$47$ $$T^{6} - 6 T^{5} + 72 T^{4} + 168 T^{3} + \cdots + 576$$
$53$ $$T^{6} + 3 T^{5} + 9 T^{4} + 6 T^{3} + \cdots + 9$$
$59$ $$T^{6} + 3 T^{5} + 225 T^{4} + \cdots + 239121$$
$61$ $$T^{6} + 15 T^{5} + 297 T^{4} + \cdots + 1014049$$
$67$ $$T^{6} - 9 T^{5} + 129 T^{4} + \cdots + 5329$$
$71$ $$(T^{3} + 21 T^{2} + 54 T - 597)^{2}$$
$73$ $$T^{6} + 129 T^{4} - 142 T^{3} + \cdots + 5041$$
$79$ $$T^{6} - 24 T^{5} + 477 T^{4} + \cdots + 271441$$
$83$ $$(T^{3} - 12 T^{2} + 45 T - 53)^{2}$$
$89$ $$T^{6} - 15 T^{5} + 189 T^{4} + \cdots + 7921$$
$97$ $$(T^{3} - 21 T^{2} + 84 T + 269)^{2}$$