# Properties

 Label 189.2.o.a Level $189$ Weight $2$ Character orbit 189.o Analytic conductor $1.509$ 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] = [189,2,Mod(62,189)]

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([1, 3]))

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

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

chi := DirichletCharacter("189.62");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.o (of order $$6$$, 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: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9$$ x^12 - 7*x^10 + 37*x^8 - 78*x^6 + 123*x^4 - 36*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} - \beta_{3} + \beta_1) q^{2} + ( - \beta_{8} - \beta_{5} - \beta_{3}) q^{4} + \beta_{10} q^{5} + (\beta_{9} - \beta_{8} + \cdots + \beta_{3}) q^{7}+ \cdots + ( - 2 \beta_{8} - \beta_{5} - \beta_{4} + \cdots + 1) q^{8}+O(q^{10})$$ q + (-b5 - b3 + b1) * q^2 + (-b8 - b5 - b3) * q^4 + b10 * q^5 + (b9 - b8 - b4 + b3) * q^7 + (-2*b8 - b5 - b4 + b3 - b1 + 1) * q^8 $$q + ( - \beta_{5} - \beta_{3} + \beta_1) q^{2} + ( - \beta_{8} - \beta_{5} - \beta_{3}) q^{4} + \beta_{10} q^{5} + (\beta_{9} - \beta_{8} + \cdots + \beta_{3}) q^{7}+ \cdots + ( - 4 \beta_{8} - 2 \beta_{7} + \cdots + 4) q^{98}+O(q^{100})$$ q + (-b5 - b3 + b1) * q^2 + (-b8 - b5 - b3) * q^4 + b10 * q^5 + (b9 - b8 - b4 + b3) * q^7 + (-2*b8 - b5 - b4 + b3 - b1 + 1) * q^8 + (-b11 - b9 + b6 + b2) * q^10 + (b8 + b5 - b4 + b3) * q^11 + (-b10 - 2*b7 - b2) * q^13 + (-b10 + 2*b9 + b5 - b2 - b1 + 1) * q^14 + (b8 + b4 - b3) * q^16 + (b11 + b6) * q^17 + (b11 + 2*b10 - b9 + b7 - b6 + b2) * q^19 + (-b9 + 2*b2) * q^20 + (2*b8 + 3*b5 + 2*b3 - 1) * q^22 + (-b5 - b1 - 1) * q^23 + (b3 + b1) * q^25 + (-b11 - 2*b9 - b7 - b6 - 2*b2) * q^26 + (b11 - 2*b10 + 2*b9 - b7 - b6 + b4 - 2*b2 - b1) * q^28 + (b5 - b3 + b1 - 4) * q^29 + (b10 + 2*b7 + 2*b6) * q^31 + (2*b8 - b5 + 4*b4 - 2*b3 + b1 + 1) * q^32 + (-b10 + b7) * q^34 + (-b11 + 2*b8 + b7 - b6 - b5 + b4 + b3 + b1 + 1) * q^35 + (-b4 + b1 - 1) * q^37 + (-b11 - 2*b9 + 2*b6 + 4*b2) * q^38 + (b10 + 2*b7 - 2*b6 + b2) * q^40 + (-2*b10 + 2*b9 - b2) * q^41 + (b8 - 2*b5 + b4 - 3*b3 - 2*b1) * q^43 + (7*b5 + 3*b3 - 2) * q^44 + (2*b5 - b4 + 2*b3 - 3*b1 - 2) * q^46 + (b11 + b10 + b9 + b7 - 2*b6 - 2*b2) * q^47 + (-2*b6 - 3*b5 - 4*b3 + 2*b1 - 1) * q^49 + (b8 + b5 + 2*b4 - b3 + b1 + 2) * q^50 + (-b11 + b10 - 5*b9 - b7) * q^52 + (-4*b8 + b5 - 2*b4 - b3 - 2*b1 - 1) * q^53 + (-b11 + 2*b9 + b6 - 2*b2) * q^55 + (b11 - b10 + b9 - b7 - 2*b6 + b5 - 2*b3 - 2*b2 + 2*b1 - 6) * q^56 + (-b8 + b5 + 3*b3 - 2*b1 + 2) * q^58 + (2*b11 + b10 - 4*b9 - b6 + 2*b2) * q^59 + (b11 + b9) * q^61 + (b11 + b9 + 2*b7 + b6 + b2) * q^62 + (-4*b5 + b4 - 4*b3 + 7*b1 - 2) * q^64 + (-b8 - 4*b5 + b4 + b3 - 2*b1 + 10) * q^65 + (3*b8 - 2*b5 - b3 + 2*b1 + 1) * q^67 + (-2*b11 + 2*b9 + b6 - b2) * q^68 + (b11 + b10 + b9 + 3*b8 - b7 + 2*b5 + 3*b4 - b3 + 2*b1) * q^70 + (4*b8 - 7*b5 + 2*b4 - 3*b3 + 2*b1 + 2) * q^71 + (-3*b11 - 2*b10 - 3*b9 - b7 + 3*b6 + 3*b2) * q^73 + (b3 - b1 + 2) * q^74 + (b10 + 2*b7 + 2*b6 + 4*b2) * q^76 + (-2*b11 + 2*b10 - 2*b9 - b8 + b6 + 2*b5 - 2*b4 + b3 + b2 + 1) * q^77 + (-3*b8 - 3*b4 + 2*b3 - b1) * q^79 + (b11 - b7 + b6) * q^80 + (2*b11 - 2*b10 + 5*b9 - b7 - 2*b6 - 5*b2) * q^82 + (-b11 - 2*b10 + 2*b9 - 2*b7 + 2*b6 - 4*b2) * q^83 + (-6*b8 + 3*b5 + 6*b3 - 6*b1 + 3) * q^85 + (-2*b8 - 3*b5 - 4*b4 + 2*b3 - 3*b1 - 5) * q^86 + (-b8 + 6*b5 - b4 + 2*b3 + b1) * q^88 + (3*b9 + 3*b2) * q^89 + (2*b11 + 2*b10 + b7 - 2*b6 + 4*b5 - 3*b4 + 4*b3 - 5*b1) * q^91 + (2*b8 + 5*b5 - 2*b4 + 4*b3 - 2*b1 - 2) * q^92 + (-2*b10 - 4*b7 + b6 - 2*b2) * q^94 + (-3*b8 - 3*b5 - 6*b4 + 3*b3 + 3*b1 - 6) * q^95 + (-b11 + b10 - b7) * q^97 + (-4*b8 - 2*b7 - 9*b5 - 2*b4 - b3 - 2*b1 + 4) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{2} + 2 q^{4} - 2 q^{7}+O(q^{10})$$ 12 * q + 6 * q^2 + 2 * q^4 - 2 * q^7 $$12 q + 6 q^{2} + 2 q^{4} - 2 q^{7} + 12 q^{14} + 2 q^{16} - 10 q^{22} - 24 q^{23} - 8 q^{28} - 30 q^{29} + 12 q^{32} - 4 q^{37} - 10 q^{43} - 40 q^{46} + 6 q^{49} + 36 q^{50} - 42 q^{56} + 2 q^{58} + 16 q^{64} + 78 q^{65} + 12 q^{67} + 18 q^{70} + 12 q^{74} + 24 q^{77} - 6 q^{79} - 6 q^{85} - 96 q^{86} + 34 q^{88} - 24 q^{91} - 30 q^{92} - 72 q^{95}+O(q^{100})$$ 12 * q + 6 * q^2 + 2 * q^4 - 2 * q^7 + 12 * q^14 + 2 * q^16 - 10 * q^22 - 24 * q^23 - 8 * q^28 - 30 * q^29 + 12 * q^32 - 4 * q^37 - 10 * q^43 - 40 * q^46 + 6 * q^49 + 36 * q^50 - 42 * q^56 + 2 * q^58 + 16 * q^64 + 78 * q^65 + 12 * q^67 + 18 * q^70 + 12 * q^74 + 24 * q^77 - 6 * q^79 - 6 * q^85 - 96 * q^86 + 34 * q^88 - 24 * q^91 - 30 * q^92 - 72 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( -28\nu^{10} + 148\nu^{8} + 446\nu^{6} - 3807\nu^{4} + 17052\nu^{2} + 4446 ) / 12897$$ (-28*v^10 + 148*v^8 + 446*v^6 - 3807*v^4 + 17052*v^2 + 4446) / 12897 $$\beta_{2}$$ $$=$$ $$( -49\nu^{11} + 259\nu^{9} - 1369\nu^{7} + 861\nu^{5} - 252\nu^{3} - 15864\nu ) / 4299$$ (-49*v^11 + 259*v^9 - 1369*v^7 + 861*v^5 - 252*v^3 - 15864*v) / 4299 $$\beta_{3}$$ $$=$$ $$( -164\nu^{10} + 1481\nu^{8} - 7214\nu^{6} + 17007\nu^{4} - 16197\nu^{2} - 3438 ) / 12897$$ (-164*v^10 + 1481*v^8 - 7214*v^6 + 17007*v^4 - 16197*v^2 - 3438) / 12897 $$\beta_{4}$$ $$=$$ $$( -175\nu^{10} + 925\nu^{8} - 3661\nu^{6} - 1224\nu^{4} + 16296\nu^{2} - 30249 ) / 12897$$ (-175*v^10 + 925*v^8 - 3661*v^6 - 1224*v^4 + 16296*v^2 - 30249) / 12897 $$\beta_{5}$$ $$=$$ $$( 148\nu^{10} - 987\nu^{8} + 5217\nu^{6} - 10175\nu^{4} + 17343\nu^{2} - 777 ) / 4299$$ (148*v^10 - 987*v^8 + 5217*v^6 - 10175*v^4 + 17343*v^2 - 777) / 4299 $$\beta_{6}$$ $$=$$ $$( 70\nu^{11} - 370\nu^{9} + 1751\nu^{7} - 1230\nu^{5} + 360\nu^{3} + 8947\nu ) / 1433$$ (70*v^11 - 370*v^9 + 1751*v^7 - 1230*v^5 + 360*v^3 + 8947*v) / 1433 $$\beta_{7}$$ $$=$$ $$( -730\nu^{11} + 5701\nu^{9} - 30748\nu^{7} + 76698\nu^{5} - 122898\nu^{3} + 66168\nu ) / 12897$$ (-730*v^11 + 5701*v^9 - 30748*v^7 + 76698*v^5 - 122898*v^3 + 66168*v) / 12897 $$\beta_{8}$$ $$=$$ $$( -877\nu^{10} + 6478\nu^{8} - 34855\nu^{6} + 79281\nu^{4} - 123654\nu^{2} + 31473 ) / 12897$$ (-877*v^10 + 6478*v^8 - 34855*v^6 + 79281*v^4 - 123654*v^2 + 31473) / 12897 $$\beta_{9}$$ $$=$$ $$( 543\nu^{11} - 3689\nu^{9} + 19499\nu^{7} - 39839\nu^{5} + 64821\nu^{3} - 18972\nu ) / 4299$$ (543*v^11 - 3689*v^9 + 19499*v^7 - 39839*v^5 + 64821*v^3 - 18972*v) / 4299 $$\beta_{10}$$ $$=$$ $$( 2237\nu^{11} - 15509\nu^{9} + 81362\nu^{7} - 167049\nu^{5} + 249792\nu^{3} - 42912\nu ) / 12897$$ (2237*v^11 - 15509*v^9 + 81362*v^7 - 167049*v^5 + 249792*v^3 - 42912*v) / 12897 $$\beta_{11}$$ $$=$$ $$( -890\nu^{11} + 6342\nu^{9} - 33522\nu^{7} + 71935\nu^{5} - 111438\nu^{3} + 32616\nu ) / 4299$$ (-890*v^11 + 6342*v^9 - 33522*v^7 + 71935*v^5 - 111438*v^3 + 32616*v) / 4299
 $$\nu$$ $$=$$ $$( -\beta_{10} - 2\beta_{7} + \beta_{6} - \beta_{2} ) / 3$$ (-b10 - 2*b7 + b6 - b2) / 3 $$\nu^{2}$$ $$=$$ $$\beta_{8} + 2\beta_{5} + \beta_{4} - \beta_{3}$$ b8 + 2*b5 + b4 - b3 $$\nu^{3}$$ $$=$$ $$( -4\beta_{11} - 8\beta_{10} + \beta_{9} - 4\beta_{7} + 4\beta_{6} - \beta_{2} ) / 3$$ (-4*b11 - 8*b10 + b9 - 4*b7 + 4*b6 - b2) / 3 $$\nu^{4}$$ $$=$$ $$4\beta_{8} + 6\beta_{5} - 6\beta_{3} + 5\beta _1 - 12$$ 4*b8 + 6*b5 - 6*b3 + 5*b1 - 12 $$\nu^{5}$$ $$=$$ $$( -19\beta_{11} - 16\beta_{10} - 2\beta_{9} + 16\beta_{7} ) / 3$$ (-19*b11 - 16*b10 - 2*b9 + 16*b7) / 3 $$\nu^{6}$$ $$=$$ $$-7\beta_{5} - 16\beta_{4} - 7\beta_{3} + 30\beta _1 - 51$$ -7*b5 - 16*b4 - 7*b3 + 30*b1 - 51 $$\nu^{7}$$ $$=$$ $$( 67\beta_{10} + 134\beta_{7} - 88\beta_{6} - 23\beta_{2} ) / 3$$ (67*b10 + 134*b7 - 88*b6 - 23*b2) / 3 $$\nu^{8}$$ $$=$$ $$-67\beta_{8} - 118\beta_{5} - 67\beta_{4} + 104\beta_{3} + 37\beta_1$$ -67*b8 - 118*b5 - 67*b4 + 104*b3 + 37*b1 $$\nu^{9}$$ $$=$$ $$( 400\beta_{11} + 578\beta_{10} + 134\beta_{9} + 289\beta_{7} - 400\beta_{6} - 134\beta_{2} ) / 3$$ (400*b11 + 578*b10 + 134*b9 + 289*b7 - 400*b6 - 134*b2) / 3 $$\nu^{10}$$ $$=$$ $$-289\beta_{8} - 333\beta_{5} + 645\beta_{3} - 467\beta _1 + 978$$ -289*b8 - 333*b5 + 645*b3 - 467*b1 + 978 $$\nu^{11}$$ $$=$$ $$( 1801\beta_{11} + 1267\beta_{10} + 668\beta_{9} - 1267\beta_{7} ) / 3$$ (1801*b11 + 1267*b10 + 668*b9 - 1267*b7) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$1 - \beta_{5}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
62.1
 −1.82904 + 1.05600i 1.82904 − 1.05600i 0.474636 − 0.274031i −0.474636 + 0.274031i −1.29589 + 0.748185i 1.29589 − 0.748185i −1.82904 − 1.05600i 1.82904 + 1.05600i 0.474636 + 0.274031i −0.474636 − 0.274031i −1.29589 − 0.748185i 1.29589 + 0.748185i
−1.02704 0.592963i 0 −0.296790 0.514055i −1.41899 2.45776i 0 −2.07253 + 1.64457i 3.07579i 0 3.36562i
62.2 −1.02704 0.592963i 0 −0.296790 0.514055i 1.41899 + 2.45776i 0 −0.387972 + 2.61715i 3.07579i 0 3.36562i
62.3 0.555632 + 0.320794i 0 −0.794182 1.37556i −1.10552 1.91482i 0 −0.906161 2.48573i 2.30225i 0 1.41858i
62.4 0.555632 + 0.320794i 0 −0.794182 1.37556i 1.10552 + 1.91482i 0 2.60579 0.458109i 2.30225i 0 1.41858i
62.5 1.97141 + 1.13819i 0 1.59097 + 2.75564i −0.717144 1.24213i 0 2.16235 + 1.52455i 2.69056i 0 3.26499i
62.6 1.97141 + 1.13819i 0 1.59097 + 2.75564i 0.717144 + 1.24213i 0 −2.40147 1.11037i 2.69056i 0 3.26499i
125.1 −1.02704 + 0.592963i 0 −0.296790 + 0.514055i −1.41899 + 2.45776i 0 −2.07253 1.64457i 3.07579i 0 3.36562i
125.2 −1.02704 + 0.592963i 0 −0.296790 + 0.514055i 1.41899 2.45776i 0 −0.387972 2.61715i 3.07579i 0 3.36562i
125.3 0.555632 0.320794i 0 −0.794182 + 1.37556i −1.10552 + 1.91482i 0 −0.906161 + 2.48573i 2.30225i 0 1.41858i
125.4 0.555632 0.320794i 0 −0.794182 + 1.37556i 1.10552 1.91482i 0 2.60579 + 0.458109i 2.30225i 0 1.41858i
125.5 1.97141 1.13819i 0 1.59097 2.75564i −0.717144 + 1.24213i 0 2.16235 1.52455i 2.69056i 0 3.26499i
125.6 1.97141 1.13819i 0 1.59097 2.75564i 0.717144 1.24213i 0 −2.40147 + 1.11037i 2.69056i 0 3.26499i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 62.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.o.a 12
3.b odd 2 1 63.2.o.a 12
4.b odd 2 1 3024.2.cc.a 12
7.b odd 2 1 inner 189.2.o.a 12
7.c even 3 1 1323.2.i.c 12
7.c even 3 1 1323.2.s.c 12
7.d odd 6 1 1323.2.i.c 12
7.d odd 6 1 1323.2.s.c 12
9.c even 3 1 63.2.o.a 12
9.c even 3 1 567.2.c.c 12
9.d odd 6 1 inner 189.2.o.a 12
9.d odd 6 1 567.2.c.c 12
12.b even 2 1 1008.2.cc.a 12
21.c even 2 1 63.2.o.a 12
21.g even 6 1 441.2.i.c 12
21.g even 6 1 441.2.s.c 12
21.h odd 6 1 441.2.i.c 12
21.h odd 6 1 441.2.s.c 12
28.d even 2 1 3024.2.cc.a 12
36.f odd 6 1 1008.2.cc.a 12
36.h even 6 1 3024.2.cc.a 12
63.g even 3 1 441.2.i.c 12
63.h even 3 1 441.2.s.c 12
63.i even 6 1 1323.2.s.c 12
63.j odd 6 1 1323.2.s.c 12
63.k odd 6 1 441.2.i.c 12
63.l odd 6 1 63.2.o.a 12
63.l odd 6 1 567.2.c.c 12
63.n odd 6 1 1323.2.i.c 12
63.o even 6 1 inner 189.2.o.a 12
63.o even 6 1 567.2.c.c 12
63.s even 6 1 1323.2.i.c 12
63.t odd 6 1 441.2.s.c 12
84.h odd 2 1 1008.2.cc.a 12
252.s odd 6 1 3024.2.cc.a 12
252.bi even 6 1 1008.2.cc.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.o.a 12 3.b odd 2 1
63.2.o.a 12 9.c even 3 1
63.2.o.a 12 21.c even 2 1
63.2.o.a 12 63.l odd 6 1
189.2.o.a 12 1.a even 1 1 trivial
189.2.o.a 12 7.b odd 2 1 inner
189.2.o.a 12 9.d odd 6 1 inner
189.2.o.a 12 63.o even 6 1 inner
441.2.i.c 12 21.g even 6 1
441.2.i.c 12 21.h odd 6 1
441.2.i.c 12 63.g even 3 1
441.2.i.c 12 63.k odd 6 1
441.2.s.c 12 21.g even 6 1
441.2.s.c 12 21.h odd 6 1
441.2.s.c 12 63.h even 3 1
441.2.s.c 12 63.t odd 6 1
567.2.c.c 12 9.c even 3 1
567.2.c.c 12 9.d odd 6 1
567.2.c.c 12 63.l odd 6 1
567.2.c.c 12 63.o even 6 1
1008.2.cc.a 12 12.b even 2 1
1008.2.cc.a 12 36.f odd 6 1
1008.2.cc.a 12 84.h odd 2 1
1008.2.cc.a 12 252.bi even 6 1
1323.2.i.c 12 7.c even 3 1
1323.2.i.c 12 7.d odd 6 1
1323.2.i.c 12 63.n odd 6 1
1323.2.i.c 12 63.s even 6 1
1323.2.s.c 12 7.c even 3 1
1323.2.s.c 12 7.d odd 6 1
1323.2.s.c 12 63.i even 6 1
1323.2.s.c 12 63.j odd 6 1
3024.2.cc.a 12 4.b odd 2 1
3024.2.cc.a 12 28.d even 2 1
3024.2.cc.a 12 36.h even 6 1
3024.2.cc.a 12 252.s odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} - 3 T^{5} + T^{4} + \cdots + 3)^{2}$$
$3$ $$T^{12}$$
$5$ $$T^{12} + 15 T^{10} + \cdots + 6561$$
$7$ $$T^{12} + 2 T^{11} + \cdots + 117649$$
$11$ $$(T^{6} - 11 T^{4} + 121 T^{2} + \cdots + 3)^{2}$$
$13$ $$T^{12} - 45 T^{10} + \cdots + 1750329$$
$17$ $$(T^{6} - 54 T^{4} + \cdots - 729)^{2}$$
$19$ $$(T^{6} + 63 T^{4} + \cdots + 2187)^{2}$$
$23$ $$(T^{6} + 12 T^{5} + \cdots + 27)^{2}$$
$29$ $$(T^{6} + 15 T^{5} + \cdots + 243)^{2}$$
$31$ $$T^{12} - 129 T^{10} + \cdots + 729$$
$37$ $$(T^{3} + T^{2} - 4 T - 1)^{4}$$
$41$ $$T^{12} + 72 T^{10} + \cdots + 531441$$
$43$ $$(T^{6} + 5 T^{5} + \cdots + 38809)^{2}$$
$47$ $$T^{12} + 93 T^{10} + \cdots + 6561$$
$53$ $$(T^{6} + 118 T^{4} + \cdots + 20667)^{2}$$
$59$ $$T^{12} + \cdots + 18539817921$$
$61$ $$T^{12} - 24 T^{10} + \cdots + 59049$$
$67$ $$(T^{6} - 6 T^{5} + 51 T^{4} + \cdots + 49)^{2}$$
$71$ $$(T^{6} + 163 T^{4} + \cdots + 363)^{2}$$
$73$ $$(T^{6} + 279 T^{4} + \cdots + 2187)^{2}$$
$79$ $$(T^{6} + 3 T^{5} + \cdots + 6241)^{2}$$
$83$ $$T^{12} + \cdots + 132211504881$$
$89$ $$(T^{6} - 324 T^{4} + \cdots - 531441)^{2}$$
$97$ $$T^{12} - 69 T^{10} + \cdots + 10673289$$