Properties

 Label 243.2.c.e Level $243$ Weight $2$ Character orbit 243.c Analytic conductor $1.940$ 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] = [243,2,Mod(82,243)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(243, 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("243.82");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$243 = 3^{5}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 243.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: $$1.94036476912$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + \beta_1 - 1) q^{2} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1) q^{5} + ( - \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{4} + \beta_{3} + 2) q^{8}+O(q^{10})$$ q + (b5 + b1 - 1) * q^2 + (-2*b5 + 2*b4 + 2*b3 - b2 - b1) * q^4 + (b5 - b4 - b3 - 2*b1) * q^5 + (-b5 + 2*b3 - 2*b2 - b1 + 1) * q^7 + (-2*b4 + b3 + 2) * q^8 $$q + (\beta_{5} + \beta_1 - 1) q^{2} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1) q^{5} + ( - \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{4} + \beta_{3} + 2) q^{8} + ( - \beta_{4} - 2 \beta_{3}) q^{10} + ( - 2 \beta_{5} + 3 \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{11} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{13} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + \beta_1) q^{14} + (3 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{16} + 3 q^{17} + (3 \beta_{3} - 1) q^{19} + (2 \beta_{5} - \beta_1 + 1) q^{20} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 5 \beta_{2}) q^{22} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{23} + ( - 4 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{25} + (\beta_{4} + \beta_{3} - 4) q^{26} + (6 \beta_{4} + 3 \beta_{3} - 4) q^{28} + (\beta_{5} - 3 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 4) q^{29} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{31} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3}) q^{32} + (3 \beta_{5} + 3 \beta_1 - 3) q^{34} + ( - \beta_{4} - 4 \beta_{3} - 2) q^{35} + ( - 3 \beta_{4} - 3 \beta_{3} - 1) q^{37} + (2 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 4) q^{38} + (\beta_{5} - \beta_{4} - \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{40} + ( - 5 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + \beta_1) q^{41} + (2 \beta_{5} - \beta_{3} + \beta_{2} - 4 \beta_1 + 4) q^{43} + (5 \beta_{4} + 2 \beta_{3} - 5) q^{44} + ( - 7 \beta_{4} - 2 \beta_{3} + 3) q^{46} + ( - 5 \beta_{5} - 2 \beta_1 + 2) q^{47} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{49} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 8 \beta_1) q^{50} + ( - 3 \beta_{5} + 3 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 4) q^{52} + ( - 3 \beta_{3} + 6) q^{53} + (2 \beta_{4} - 2 \beta_{3} + 3) q^{55} + ( - 5 \beta_{5} + 3 \beta_{3} - 3 \beta_{2} - 11 \beta_1 + 11) q^{56} + ( - 8 \beta_{5} + 8 \beta_{4} + 8 \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{58} + (\beta_{5} - \beta_{4} - \beta_{3} + 7 \beta_1) q^{59} + ( - \beta_{5} + 5 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 2) q^{61} + (7 \beta_{4} + 4 \beta_{3} - 4) q^{62} + (3 \beta_{4} + 4) q^{64} + ( - 2 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{65} + ( - 5 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - \beta_{2} - 2 \beta_1) q^{67} + ( - 6 \beta_{5} + 6 \beta_{4} + 6 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{68} + ( - 4 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{70} + ( - 9 \beta_{4} - 6 \beta_{3} - 3) q^{71} + (6 \beta_{4} + 3 \beta_{3} + 2) q^{73} + (2 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 5) q^{74} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{76} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 8 \beta_1) q^{77} + (5 \beta_{5} - 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{79} + ( - 2 \beta_{4} + \beta_{3} - 1) q^{80} + ( - 7 \beta_{4} + \beta_{3} + 6) q^{82} + (\beta_{5} + 3 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 2) q^{83} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 6 \beta_1) q^{85} + (\beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_1) q^{86} + ( - 5 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} - 12 \beta_1 + 12) q^{88} + (9 \beta_{4} + 3 \beta_{3}) q^{89} + ( - 5 \beta_{4} - \beta_{3} - 2) q^{91} + (11 \beta_{5} - 6 \beta_{3} + 6 \beta_{2} + 8 \beta_1 - 8) q^{92} + (7 \beta_{5} - 7 \beta_{4} - 7 \beta_{3} + 5 \beta_{2} + 12 \beta_1) q^{94} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 6 \beta_{2} - \beta_1) q^{95} + (8 \beta_{5} - 4 \beta_{3} + 4 \beta_{2} + 5 \beta_1 - 5) q^{97} + ( - 3 \beta_{4} + 3) q^{98}+O(q^{100})$$ q + (b5 + b1 - 1) * q^2 + (-2*b5 + 2*b4 + 2*b3 - b2 - b1) * q^4 + (b5 - b4 - b3 - 2*b1) * q^5 + (-b5 + 2*b3 - 2*b2 - b1 + 1) * q^7 + (-2*b4 + b3 + 2) * q^8 + (-b4 - 2*b3) * q^10 + (-2*b5 + 3*b3 - 3*b2 + b1 - 1) * q^11 + (b5 - b4 - b3 - b2 + b1) * q^13 + (4*b5 - 4*b4 - 4*b3 + 3*b2 + b1) * q^14 + (3*b5 - 3*b3 + 3*b2 + b1 - 1) * q^16 + 3 * q^17 + (3*b3 - 1) * q^19 + (2*b5 - b1 + 1) * q^20 + (4*b5 - 4*b4 - 4*b3 + 5*b2) * q^22 + (-2*b5 + 2*b4 + 2*b3 - 3*b2 - 2*b1) * q^23 + (-4*b5 - b3 + b2 + b1 - 1) * q^25 + (b4 + b3 - 4) * q^26 + (6*b4 + 3*b3 - 4) * q^28 + (b5 - 3*b3 + 3*b2 + 4*b1 - 4) * q^29 + (b5 - b4 - b3 + 2*b2 + 4*b1) * q^31 + (-3*b5 + 3*b4 + 3*b3) * q^32 + (3*b5 + 3*b1 - 3) * q^34 + (-b4 - 4*b3 - 2) * q^35 + (-3*b4 - 3*b3 - 1) * q^37 + (2*b5 - 3*b3 + 3*b2 - 4*b1 + 4) * q^38 + (b5 - b4 - b3 - 4*b2 - 3*b1) * q^40 + (-5*b5 + 5*b4 + 5*b3 - 3*b2 + b1) * q^41 + (2*b5 - b3 + b2 - 4*b1 + 4) * q^43 + (5*b4 + 2*b3 - 5) * q^44 + (-7*b4 - 2*b3 + 3) * q^46 + (-5*b5 - 2*b1 + 2) * q^47 + (-2*b5 + 2*b4 + 2*b3 - b2) * q^49 + (2*b5 - 2*b4 - 2*b3 + 3*b2 + 8*b1) * q^50 + (-3*b5 + 3*b3 - 3*b2 - 4*b1 + 4) * q^52 + (-3*b3 + 6) * q^53 + (2*b4 - 2*b3 + 3) * q^55 + (-5*b5 + 3*b3 - 3*b2 - 11*b1 + 11) * q^56 + (-8*b5 + 8*b4 + 8*b3 - 4*b2 - 3*b1) * q^58 + (b5 - b4 - b3 + 7*b1) * q^59 + (-b5 + 5*b3 - 5*b2 + 2*b1 - 2) * q^61 + (7*b4 + 4*b3 - 4) * q^62 + (3*b4 + 4) * q^64 + (-2*b5 - 3*b3 + 3*b2 + b1 - 1) * q^65 + (-5*b5 + 5*b4 + 5*b3 - b2 - 2*b1) * q^67 + (-6*b5 + 6*b4 + 6*b3 - 3*b2 - 3*b1) * q^68 + (-4*b5 + 2*b3 - 2*b2 + 3*b1 - 3) * q^70 + (-9*b4 - 6*b3 - 3) * q^71 + (6*b4 + 3*b3 + 2) * q^73 + (2*b5 - 3*b3 + 3*b2 + 5*b1 - 5) * q^74 + (-b5 + b4 + b3 + b2 + b1) * q^76 + (-2*b5 + 2*b4 + 2*b3 + 3*b2 - 8*b1) * q^77 + (5*b5 - 4*b3 + 4*b2 + 2*b1 - 2) * q^79 + (-2*b4 + b3 - 1) * q^80 + (-7*b4 + b3 + 6) * q^82 + (b5 + 3*b3 - 3*b2 - 2*b1 + 2) * q^83 + (3*b5 - 3*b4 - 3*b3 - 6*b1) * q^85 + (b5 - b4 - b3 - 3*b2 + b1) * q^86 + (-5*b5 - 2*b3 + 2*b2 - 12*b1 + 12) * q^88 + (9*b4 + 3*b3) * q^89 + (-5*b4 - b3 - 2) * q^91 + (11*b5 - 6*b3 + 6*b2 + 8*b1 - 8) * q^92 + (7*b5 - 7*b4 - 7*b3 + 5*b2 + 12*b1) * q^94 + (2*b5 - 2*b4 - 2*b3 - 6*b2 - b1) * q^95 + (8*b5 - 4*b3 + 4*b2 + 5*b1 - 5) * q^97 + (-3*b4 + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 3 q^{7} + 12 q^{8}+O(q^{10})$$ 6 * q - 3 * q^2 - 3 * q^4 - 6 * q^5 + 3 * q^7 + 12 * q^8 $$6 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 3 q^{7} + 12 q^{8} - 3 q^{11} + 3 q^{13} + 3 q^{14} - 3 q^{16} + 18 q^{17} - 6 q^{19} + 3 q^{20} - 6 q^{23} - 3 q^{25} - 24 q^{26} - 24 q^{28} - 12 q^{29} + 12 q^{31} - 9 q^{34} - 12 q^{35} - 6 q^{37} + 12 q^{38} - 9 q^{40} + 3 q^{41} + 12 q^{43} - 30 q^{44} + 18 q^{46} + 6 q^{47} + 24 q^{50} + 12 q^{52} + 36 q^{53} + 18 q^{55} + 33 q^{56} - 9 q^{58} + 21 q^{59} - 6 q^{61} - 24 q^{62} + 24 q^{64} - 3 q^{65} - 6 q^{67} - 9 q^{68} - 9 q^{70} - 18 q^{71} + 12 q^{73} - 15 q^{74} + 3 q^{76} - 24 q^{77} - 6 q^{79} - 6 q^{80} + 36 q^{82} + 6 q^{83} - 18 q^{85} + 3 q^{86} + 36 q^{88} - 12 q^{91} - 24 q^{92} + 36 q^{94} - 3 q^{95} - 15 q^{97} + 18 q^{98}+O(q^{100})$$ 6 * q - 3 * q^2 - 3 * q^4 - 6 * q^5 + 3 * q^7 + 12 * q^8 - 3 * q^11 + 3 * q^13 + 3 * q^14 - 3 * q^16 + 18 * q^17 - 6 * q^19 + 3 * q^20 - 6 * q^23 - 3 * q^25 - 24 * q^26 - 24 * q^28 - 12 * q^29 + 12 * q^31 - 9 * q^34 - 12 * q^35 - 6 * q^37 + 12 * q^38 - 9 * q^40 + 3 * q^41 + 12 * q^43 - 30 * q^44 + 18 * q^46 + 6 * q^47 + 24 * q^50 + 12 * q^52 + 36 * q^53 + 18 * q^55 + 33 * q^56 - 9 * q^58 + 21 * q^59 - 6 * q^61 - 24 * q^62 + 24 * q^64 - 3 * q^65 - 6 * q^67 - 9 * q^68 - 9 * q^70 - 18 * q^71 + 12 * q^73 - 15 * q^74 + 3 * q^76 - 24 * q^77 - 6 * q^79 - 6 * q^80 + 36 * q^82 + 6 * q^83 - 18 * q^85 + 3 * q^86 + 36 * q^88 - 12 * q^91 - 24 * q^92 + 36 * q^94 - 3 * q^95 - 15 * q^97 + 18 * q^98

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{18}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$\zeta_{18}^{5} + \zeta_{18}$$ v^5 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18}$$ -v^4 + v^2 + v $$\beta_{4}$$ $$=$$ $$-\zeta_{18}^{5} + \zeta_{18}^{4}$$ -v^5 + v^4 $$\beta_{5}$$ $$=$$ $$-\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}$$ -v^5 - v^4 + v
 $$\zeta_{18}$$ $$=$$ $$( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3$$ (b5 + b4 + 2*b2) / 3 $$\zeta_{18}^{2}$$ $$=$$ $$( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3$$ (-2*b5 + b4 + 3*b3 - b2) / 3 $$\zeta_{18}^{3}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{18}^{4}$$ $$=$$ $$( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3$$ (-b5 + 2*b4 + b2) / 3 $$\zeta_{18}^{5}$$ $$=$$ $$( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3$$ (-b5 - b4 + b2) / 3

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/243\mathbb{Z}\right)^\times$$.

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
82.1
 −0.766044 − 0.642788i −0.173648 + 0.984808i 0.939693 − 0.342020i −0.766044 + 0.642788i −0.173648 − 0.984808i 0.939693 + 0.342020i
−1.26604 2.19285i 0 −2.20574 + 3.82045i −0.233956 + 0.405223i 0 1.61334 + 2.79439i 6.10607 0 1.18479
82.2 −0.673648 1.16679i 0 0.0923963 0.160035i −0.826352 + 1.43128i 0 −1.20574 2.08840i −2.94356 0 2.22668
82.3 0.439693 + 0.761570i 0 0.613341 1.06234i −1.93969 + 3.35965i 0 1.09240 + 1.89209i 2.83750 0 −3.41147
163.1 −1.26604 + 2.19285i 0 −2.20574 3.82045i −0.233956 0.405223i 0 1.61334 2.79439i 6.10607 0 1.18479
163.2 −0.673648 + 1.16679i 0 0.0923963 + 0.160035i −0.826352 1.43128i 0 −1.20574 + 2.08840i −2.94356 0 2.22668
163.3 0.439693 0.761570i 0 0.613341 + 1.06234i −1.93969 3.35965i 0 1.09240 1.89209i 2.83750 0 −3.41147
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 163.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 243.2.c.e 6
3.b odd 2 1 243.2.c.f 6
9.c even 3 1 243.2.a.f yes 3
9.c even 3 1 inner 243.2.c.e 6
9.d odd 6 1 243.2.a.e 3
9.d odd 6 1 243.2.c.f 6
27.e even 9 2 729.2.e.b 6
27.e even 9 2 729.2.e.c 6
27.e even 9 2 729.2.e.i 6
27.f odd 18 2 729.2.e.a 6
27.f odd 18 2 729.2.e.g 6
27.f odd 18 2 729.2.e.h 6
36.f odd 6 1 3888.2.a.bk 3
36.h even 6 1 3888.2.a.bd 3
45.h odd 6 1 6075.2.a.bv 3
45.j even 6 1 6075.2.a.bq 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.e 3 9.d odd 6 1
243.2.a.f yes 3 9.c even 3 1
243.2.c.e 6 1.a even 1 1 trivial
243.2.c.e 6 9.c even 3 1 inner
243.2.c.f 6 3.b odd 2 1
243.2.c.f 6 9.d odd 6 1
729.2.e.a 6 27.f odd 18 2
729.2.e.b 6 27.e even 9 2
729.2.e.c 6 27.e even 9 2
729.2.e.g 6 27.f odd 18 2
729.2.e.h 6 27.f odd 18 2
729.2.e.i 6 27.e even 9 2
3888.2.a.bd 3 36.h even 6 1
3888.2.a.bk 3 36.f odd 6 1
6075.2.a.bq 3 45.j even 6 1
6075.2.a.bv 3 45.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(243, [\chi])$$:

 $$T_{2}^{6} + 3T_{2}^{5} + 9T_{2}^{4} + 6T_{2}^{3} + 9T_{2}^{2} + 9$$ T2^6 + 3*T2^5 + 9*T2^4 + 6*T2^3 + 9*T2^2 + 9 $$T_{7}^{6} - 3T_{7}^{5} + 15T_{7}^{4} - 16T_{7}^{3} + 87T_{7}^{2} - 102T_{7} + 289$$ T7^6 - 3*T7^5 + 15*T7^4 - 16*T7^3 + 87*T7^2 - 102*T7 + 289

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 3 T^{5} + 9 T^{4} + 6 T^{3} + \cdots + 9$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 6 T^{5} + 27 T^{4} + 48 T^{3} + \cdots + 9$$
$7$ $$T^{6} - 3 T^{5} + 15 T^{4} - 16 T^{3} + \cdots + 289$$
$11$ $$T^{6} + 3 T^{5} + 27 T^{4} - 48 T^{3} + \cdots + 9$$
$13$ $$T^{6} - 3 T^{5} + 15 T^{4} - 16 T^{3} + \cdots + 289$$
$17$ $$(T - 3)^{6}$$
$19$ $$(T^{3} + 3 T^{2} - 24 T + 1)^{2}$$
$23$ $$T^{6} + 6 T^{5} + 45 T^{4} + \cdots + 2601$$
$29$ $$T^{6} + 12 T^{5} + 117 T^{4} + \cdots + 3249$$
$31$ $$T^{6} - 12 T^{5} + 105 T^{4} + \cdots + 361$$
$37$ $$(T^{3} + 3 T^{2} - 24 T + 1)^{2}$$
$41$ $$T^{6} - 3 T^{5} + 63 T^{4} + \cdots + 47961$$
$43$ $$T^{6} - 12 T^{5} + 105 T^{4} + \cdots + 361$$
$47$ $$T^{6} - 6 T^{5} + 99 T^{4} + \cdots + 71289$$
$53$ $$(T^{3} - 18 T^{2} + 81 T - 81)^{2}$$
$59$ $$T^{6} - 21 T^{5} + 297 T^{4} + \cdots + 103041$$
$61$ $$T^{6} + 6 T^{5} + 87 T^{4} + \cdots + 2809$$
$67$ $$T^{6} + 6 T^{5} + 87 T^{4} + \cdots + 11881$$
$71$ $$(T^{3} + 9 T^{2} - 162 T - 999)^{2}$$
$73$ $$(T^{3} - 6 T^{2} - 69 T + 397)^{2}$$
$79$ $$T^{6} + 6 T^{5} + 87 T^{4} + \cdots + 2809$$
$83$ $$T^{6} - 6 T^{5} + 63 T^{4} + \cdots + 2601$$
$89$ $$(T^{3} - 189 T + 999)^{2}$$
$97$ $$T^{6} + 15 T^{5} + 294 T^{4} + \cdots + 361$$