# Properties

 Label 189.2.f.b Level $189$ Weight $2$ Character orbit 189.f Analytic conductor $1.509$ 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] = [189,2,Mod(64,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("189.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.f (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: $$1.50917259820$$ 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: no (minimal twist has level 63) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{2} + (2 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + (\beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{5} - \beta_1 q^{7} + (2 \beta_{4} - \beta_{3} - 2) q^{8}+O(q^{10})$$ q + (b5 - b4 - b3 + b1) * q^2 + (2*b5 - b3 + b2 + b1 - 1) * q^4 + (b5 - b3 + b2 - b1 + 1) * q^5 - b1 * q^7 + (2*b4 - b3 - 2) * q^8 $$q + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{2} + (2 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + (\beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{5} - \beta_1 q^{7} + (2 \beta_{4} - \beta_{3} - 2) q^{8} + (\beta_{4} - \beta_{3}) q^{10} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1) q^{11} + ( - 2 \beta_{5} + 4 \beta_{3} - 4 \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{5} - \beta_1 + 1) q^{14} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - \beta_1) q^{16} + (\beta_{3} - 2) q^{17} + ( - 2 \beta_{4} - \beta_{3} - 1) q^{19} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{20} + (4 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{22} + (\beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{23} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{25} + ( - 7 \beta_{4} - \beta_{3} + 1) q^{26} + ( - 2 \beta_{4} - \beta_{3} + 1) q^{28} + ( - 5 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{29} + ( - 3 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{31} - 3 \beta_{5} q^{32} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_1) q^{34} + ( - \beta_{4} - 1) q^{35} + (3 \beta_{4} + 6 \beta_{3} - 1) q^{37} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{38} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{40} + (\beta_{5} + \beta_{3} - \beta_{2}) q^{41} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{43} + (7 \beta_{4} + 3 \beta_{3} - 5) q^{44} + ( - 5 \beta_{4} - 4 \beta_{3}) q^{46} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \beta_1) q^{47} + (\beta_1 - 1) q^{49} + (5 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 2) q^{50} + (10 \beta_{5} - 10 \beta_{4} - 10 \beta_{3} + 5 \beta_{2} + 7 \beta_1) q^{52} + (3 \beta_{4} + \beta_{3} - 2) q^{53} + (2 \beta_{4} + \beta_{3}) q^{55} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{56} + ( - 6 \beta_{5} + 9 \beta_{3} - 9 \beta_{2} - 3 \beta_1 + 3) q^{58} + ( - 5 \beta_{5} + \beta_1 - 1) q^{59} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 2 \beta_1) q^{61} + ( - \beta_{4} - \beta_{3} + 10) q^{62} + (3 \beta_{4} + 4) q^{64} + (\beta_{5} - \beta_{4} - \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{65} + ( - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 4) q^{67} + ( - 3 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{68} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{70} + ( - 3 \beta_{4} + 3 \beta_{3} - 3) q^{71} + (4 \beta_{4} + 5 \beta_{3} - 7) q^{73} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 10 \beta_1) q^{74} + (3 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 5) q^{76} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{77} + ( - 3 \beta_{2} + 7 \beta_1) q^{79} + ( - \beta_{4} - 3 \beta_{3} + 5) q^{80} - 3 q^{82} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - \beta_{2} - 6 \beta_1) q^{83} + ( - 2 \beta_{5} + 4 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 3) q^{85} + (\beta_{5} - 2 \beta_1 + 2) q^{86} + ( - 8 \beta_{5} + 8 \beta_{4} + 8 \beta_{3} - 7 \beta_{2} - 9 \beta_1) q^{88} + (3 \beta_{4} - 4 \beta_{3} - 4) q^{89} + (2 \beta_{4} - 2 \beta_{3} - 1) q^{91} + (8 \beta_{5} - 8 \beta_{4} - 8 \beta_{3} + 2 \beta_{2} + \beta_1) q^{92} + (2 \beta_{5} - \beta_{3} + \beta_{2} - 6 \beta_1 + 6) q^{94} + ( - \beta_{5} + \beta_{3} - \beta_{2} + 4 \beta_1 - 4) q^{95} + ( - 7 \beta_{5} + 7 \beta_{4} + 7 \beta_{3} - 8 \beta_{2} + \beta_1) q^{97} + (\beta_{4} + \beta_{3} - 1) q^{98}+O(q^{100})$$ q + (b5 - b4 - b3 + b1) * q^2 + (2*b5 - b3 + b2 + b1 - 1) * q^4 + (b5 - b3 + b2 - b1 + 1) * q^5 - b1 * q^7 + (2*b4 - b3 - 2) * q^8 + (b4 - b3) * q^10 + (b5 - b4 - b3 + b2 + 2*b1) * q^11 + (-2*b5 + 4*b3 - 4*b2 - b1 + 1) * q^13 + (-b5 - b1 + 1) * q^14 + (-3*b5 + 3*b4 + 3*b3 - 3*b2 - b1) * q^16 + (b3 - 2) * q^17 + (-2*b4 - b3 - 1) * q^19 + (-b5 + b4 + b3 - b2 - 2*b1) * q^20 + (4*b5 - 2*b3 + 2*b2 + 3*b1 - 3) * q^22 + (b5 + 2*b3 - 2*b2 - 4*b1 + 4) * q^23 + (b5 - b4 - b3 + 2*b2 + 2*b1) * q^25 + (-7*b4 - b3 + 1) * q^26 + (-2*b4 - b3 + 1) * q^28 + (-5*b5 + 5*b4 + 5*b3 - 4*b2 + 3*b1) * q^29 + (-3*b5 - 3*b3 + 3*b2 - b1 + 1) * q^31 - 3*b5 * q^32 + (-b5 + b4 + b3 + b2 - 3*b1) * q^34 + (-b4 - 1) * q^35 + (3*b4 + 6*b3 - 1) * q^37 + (2*b5 - 2*b4 - 2*b3 + 3*b2 + 2*b1) * q^38 + (-2*b5 - 2*b3 + 2*b2 - 3*b1 + 3) * q^40 + (b5 + b3 - b2) * q^41 + (-b5 + b4 + b3 + b2 + b1) * q^43 + (7*b4 + 3*b3 - 5) * q^44 + (-5*b4 - 4*b3) * q^46 + (-2*b5 + 2*b4 + 2*b3 + 3*b2 + b1) * q^47 + (b1 - 1) * q^49 + (5*b5 - 3*b3 + 3*b2 + 2*b1 - 2) * q^50 + (10*b5 - 10*b4 - 10*b3 + 5*b2 + 7*b1) * q^52 + (3*b4 + b3 - 2) * q^53 + (2*b4 + b3) * q^55 + (2*b5 - 2*b4 - 2*b3 + 3*b2 + 2*b1) * q^56 + (-6*b5 + 9*b3 - 9*b2 - 3*b1 + 3) * q^58 + (-5*b5 + b1 - 1) * q^59 + (3*b5 - 3*b4 - 3*b3 - 2*b1) * q^61 + (-b4 - b3 + 10) * q^62 + (3*b4 + 4) * q^64 + (b5 - b4 - b3 - 5*b2 + 5*b1) * q^65 + (-3*b3 + 3*b2 - 4*b1 + 4) * q^67 + (-3*b5 + 2*b3 - 2*b2 - 2*b1 + 2) * q^68 + (b5 - b4 - b3 + 2*b2) * q^70 + (-3*b4 + 3*b3 - 3) * q^71 + (4*b4 + 5*b3 - 7) * q^73 + (-b5 + b4 + b3 - 10*b1) * q^74 + (3*b5 - 3*b3 + 3*b2 + 5*b1 - 5) * q^76 + (-b5 + b3 - b2 - 2*b1 + 2) * q^77 + (-3*b2 + 7*b1) * q^79 + (-b4 - 3*b3 + 5) * q^80 - 3 * q^82 + (4*b5 - 4*b4 - 4*b3 - b2 - 6*b1) * q^83 + (-2*b5 + 4*b3 - 4*b2 + 3*b1 - 3) * q^85 + (b5 - 2*b1 + 2) * q^86 + (-8*b5 + 8*b4 + 8*b3 - 7*b2 - 9*b1) * q^88 + (3*b4 - 4*b3 - 4) * q^89 + (2*b4 - 2*b3 - 1) * q^91 + (8*b5 - 8*b4 - 8*b3 + 2*b2 + b1) * q^92 + (2*b5 - b3 + b2 - 6*b1 + 6) * q^94 + (-b5 + b3 - b2 + 4*b1 - 4) * q^95 + (-7*b5 + 7*b4 + 7*b3 - 8*b2 + b1) * q^97 + (b4 + b3 - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{2} - 3 q^{4} + 3 q^{5} - 3 q^{7} - 12 q^{8}+O(q^{10})$$ 6 * q + 3 * q^2 - 3 * q^4 + 3 * q^5 - 3 * q^7 - 12 * q^8 $$6 q + 3 q^{2} - 3 q^{4} + 3 q^{5} - 3 q^{7} - 12 q^{8} + 6 q^{11} + 3 q^{13} + 3 q^{14} - 3 q^{16} - 12 q^{17} - 6 q^{19} - 6 q^{20} - 9 q^{22} + 12 q^{23} + 6 q^{25} + 6 q^{26} + 6 q^{28} + 9 q^{29} + 3 q^{31} - 9 q^{34} - 6 q^{35} - 6 q^{37} + 6 q^{38} + 9 q^{40} + 3 q^{43} - 30 q^{44} + 3 q^{47} - 3 q^{49} - 6 q^{50} + 21 q^{52} - 12 q^{53} + 6 q^{56} + 9 q^{58} - 3 q^{59} - 6 q^{61} + 60 q^{62} + 24 q^{64} + 15 q^{65} + 12 q^{67} + 6 q^{68} - 18 q^{71} - 42 q^{73} - 30 q^{74} - 15 q^{76} + 6 q^{77} + 21 q^{79} + 30 q^{80} - 18 q^{82} - 18 q^{83} - 9 q^{85} + 6 q^{86} - 27 q^{88} - 24 q^{89} - 6 q^{91} + 3 q^{92} + 18 q^{94} - 12 q^{95} + 3 q^{97} - 6 q^{98}+O(q^{100})$$ 6 * q + 3 * q^2 - 3 * q^4 + 3 * q^5 - 3 * q^7 - 12 * q^8 + 6 * q^11 + 3 * q^13 + 3 * q^14 - 3 * q^16 - 12 * q^17 - 6 * q^19 - 6 * q^20 - 9 * q^22 + 12 * q^23 + 6 * q^25 + 6 * q^26 + 6 * q^28 + 9 * q^29 + 3 * q^31 - 9 * q^34 - 6 * q^35 - 6 * q^37 + 6 * q^38 + 9 * q^40 + 3 * q^43 - 30 * q^44 + 3 * q^47 - 3 * q^49 - 6 * q^50 + 21 * q^52 - 12 * q^53 + 6 * q^56 + 9 * q^58 - 3 * q^59 - 6 * q^61 + 60 * q^62 + 24 * q^64 + 15 * q^65 + 12 * q^67 + 6 * q^68 - 18 * q^71 - 42 * q^73 - 30 * q^74 - 15 * q^76 + 6 * q^77 + 21 * q^79 + 30 * q^80 - 18 * q^82 - 18 * q^83 - 9 * q^85 + 6 * q^86 - 27 * q^88 - 24 * q^89 - 6 * q^91 + 3 * q^92 + 18 * q^94 - 12 * q^95 + 3 * q^97 - 6 * 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 $$\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$-1 + \beta_{1}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 0.939693 + 0.342020i −0.173648 − 0.984808i −0.766044 + 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i −0.766044 − 0.642788i
−0.439693 0.761570i 0 0.613341 1.06234i 0.673648 1.16679i 0 −0.500000 0.866025i −2.83750 0 −1.18479
64.2 0.673648 + 1.16679i 0 0.0923963 0.160035i 1.26604 2.19285i 0 −0.500000 0.866025i 2.94356 0 3.41147
64.3 1.26604 + 2.19285i 0 −2.20574 + 3.82045i −0.439693 + 0.761570i 0 −0.500000 0.866025i −6.10607 0 −2.22668
127.1 −0.439693 + 0.761570i 0 0.613341 + 1.06234i 0.673648 + 1.16679i 0 −0.500000 + 0.866025i −2.83750 0 −1.18479
127.2 0.673648 1.16679i 0 0.0923963 + 0.160035i 1.26604 + 2.19285i 0 −0.500000 + 0.866025i 2.94356 0 3.41147
127.3 1.26604 2.19285i 0 −2.20574 3.82045i −0.439693 0.761570i 0 −0.500000 + 0.866025i −6.10607 0 −2.22668
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.3 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 189.2.f.b 6
3.b odd 2 1 63.2.f.a 6
4.b odd 2 1 3024.2.r.k 6
7.b odd 2 1 1323.2.f.d 6
7.c even 3 1 1323.2.g.d 6
7.c even 3 1 1323.2.h.c 6
7.d odd 6 1 1323.2.g.e 6
7.d odd 6 1 1323.2.h.b 6
9.c even 3 1 inner 189.2.f.b 6
9.c even 3 1 567.2.a.c 3
9.d odd 6 1 63.2.f.a 6
9.d odd 6 1 567.2.a.h 3
12.b even 2 1 1008.2.r.h 6
21.c even 2 1 441.2.f.c 6
21.g even 6 1 441.2.g.b 6
21.g even 6 1 441.2.h.e 6
21.h odd 6 1 441.2.g.c 6
21.h odd 6 1 441.2.h.d 6
36.f odd 6 1 3024.2.r.k 6
36.f odd 6 1 9072.2.a.bs 3
36.h even 6 1 1008.2.r.h 6
36.h even 6 1 9072.2.a.ca 3
63.g even 3 1 1323.2.h.c 6
63.h even 3 1 1323.2.g.d 6
63.i even 6 1 441.2.g.b 6
63.j odd 6 1 441.2.g.c 6
63.k odd 6 1 1323.2.h.b 6
63.l odd 6 1 1323.2.f.d 6
63.l odd 6 1 3969.2.a.l 3
63.n odd 6 1 441.2.h.d 6
63.o even 6 1 441.2.f.c 6
63.o even 6 1 3969.2.a.q 3
63.s even 6 1 441.2.h.e 6
63.t odd 6 1 1323.2.g.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a 6 3.b odd 2 1
63.2.f.a 6 9.d odd 6 1
189.2.f.b 6 1.a even 1 1 trivial
189.2.f.b 6 9.c even 3 1 inner
441.2.f.c 6 21.c even 2 1
441.2.f.c 6 63.o even 6 1
441.2.g.b 6 21.g even 6 1
441.2.g.b 6 63.i even 6 1
441.2.g.c 6 21.h odd 6 1
441.2.g.c 6 63.j odd 6 1
441.2.h.d 6 21.h odd 6 1
441.2.h.d 6 63.n odd 6 1
441.2.h.e 6 21.g even 6 1
441.2.h.e 6 63.s even 6 1
567.2.a.c 3 9.c even 3 1
567.2.a.h 3 9.d odd 6 1
1008.2.r.h 6 12.b even 2 1
1008.2.r.h 6 36.h even 6 1
1323.2.f.d 6 7.b odd 2 1
1323.2.f.d 6 63.l odd 6 1
1323.2.g.d 6 7.c even 3 1
1323.2.g.d 6 63.h even 3 1
1323.2.g.e 6 7.d odd 6 1
1323.2.g.e 6 63.t odd 6 1
1323.2.h.b 6 7.d odd 6 1
1323.2.h.b 6 63.k odd 6 1
1323.2.h.c 6 7.c even 3 1
1323.2.h.c 6 63.g even 3 1
3024.2.r.k 6 4.b odd 2 1
3024.2.r.k 6 36.f odd 6 1
3969.2.a.l 3 63.l odd 6 1
3969.2.a.q 3 63.o even 6 1
9072.2.a.bs 3 36.f odd 6 1
9072.2.a.ca 3 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 3T_{2}^{5} + 9T_{2}^{4} - 6T_{2}^{3} + 9T_{2}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 3 T^{5} + 9 T^{4} - 6 T^{3} + \cdots + 9$$
$3$ $$T^{6}$$
$5$ $$T^{6} - 3 T^{5} + 9 T^{4} - 6 T^{3} + \cdots + 9$$
$7$ $$(T^{2} + T + 1)^{3}$$
$11$ $$T^{6} - 6 T^{5} + 27 T^{4} - 48 T^{3} + \cdots + 9$$
$13$ $$T^{6} - 3 T^{5} + 42 T^{4} + \cdots + 11449$$
$17$ $$(T^{3} + 6 T^{2} + 9 T + 3)^{2}$$
$19$ $$(T^{3} + 3 T^{2} - 6 T - 17)^{2}$$
$23$ $$T^{6} - 12 T^{5} + 117 T^{4} - 330 T^{3} + \cdots + 9$$
$29$ $$T^{6} - 9 T^{5} + 117 T^{4} + \cdots + 110889$$
$31$ $$T^{6} - 3 T^{5} + 87 T^{4} + \cdots + 104329$$
$37$ $$(T^{3} + 3 T^{2} - 78 T - 323)^{2}$$
$41$ $$T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81$$
$43$ $$T^{6} - 3 T^{5} + 15 T^{4} + 20 T^{3} + \cdots + 1$$
$47$ $$T^{6} - 3 T^{5} + 63 T^{4} + \cdots + 2601$$
$53$ $$(T^{3} + 6 T^{2} - 9 T + 3)^{2}$$
$59$ $$T^{6} + 3 T^{5} + 81 T^{4} + \cdots + 2601$$
$61$ $$T^{6} + 6 T^{5} + 51 T^{4} - 52 T^{3} + \cdots + 361$$
$67$ $$T^{6} - 12 T^{5} + 123 T^{4} + \cdots + 289$$
$71$ $$(T^{3} + 9 T^{2} - 54 T + 27)^{2}$$
$73$ $$(T^{3} + 21 T^{2} + 84 T - 269)^{2}$$
$79$ $$T^{6} - 21 T^{5} + 321 T^{4} + \cdots + 32761$$
$83$ $$T^{6} + 18 T^{5} + 279 T^{4} + \cdots + 81$$
$89$ $$(T^{3} + 12 T^{2} - 63 T - 813)^{2}$$
$97$ $$T^{6} - 3 T^{5} + 177 T^{4} + \cdots + 104329$$