# Properties

 Label 882.2.e.p Level $882$ Weight $2$ Character orbit 882.e Analytic conductor $7.043$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,2,Mod(373,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("882.373");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 882.e (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: $$7.04280545828$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ x^6 - 3*x^5 + 10*x^4 - 15*x^3 + 19*x^2 - 12*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{3} + q^{4} + (\beta_{5} - 2 \beta_{4} - \beta_1) q^{5} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{6} + q^{8} + (\beta_{5} - 2 \beta_{4} - 2 \beta_1 - 1) q^{9}+O(q^{10})$$ q + q^2 + (b5 + b4 - b3) * q^3 + q^4 + (b5 - 2*b4 - b1) * q^5 + (b5 + b4 - b3) * q^6 + q^8 + (b5 - 2*b4 - 2*b1 - 1) * q^9 $$q + q^{2} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{3} + q^{4} + (\beta_{5} - 2 \beta_{4} - \beta_1) q^{5} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{6} + q^{8} + (\beta_{5} - 2 \beta_{4} - 2 \beta_1 - 1) q^{9} + (\beta_{5} - 2 \beta_{4} - \beta_1) q^{10} + ( - \beta_{5} - \beta_{3} + \cdots - \beta_1) q^{11}+ \cdots + (3 \beta_{5} + 7 \beta_{4} - \beta_{3} + \cdots - 8) q^{99}+O(q^{100})$$ q + q^2 + (b5 + b4 - b3) * q^3 + q^4 + (b5 - 2*b4 - b1) * q^5 + (b5 + b4 - b3) * q^6 + q^8 + (b5 - 2*b4 - 2*b1 - 1) * q^9 + (b5 - 2*b4 - b1) * q^10 + (-b5 - b3 + b2 - b1) * q^11 + (b5 + b4 - b3) * q^12 + (-b5 + b4 + b3 + b2 + b1 + 1) * q^13 + (3*b5 - 2*b4 - b3 - 2*b2 - b1 - 2) * q^15 + q^16 + (-4*b5 - 2*b4 + 2*b3 + 2) * q^17 + (b5 - 2*b4 - 2*b1 - 1) * q^18 + (b4 - b3 - b1 + 1) * q^19 + (b5 - 2*b4 - b1) * q^20 + (-b5 - b3 + b2 - b1) * q^22 + (-2*b5 + 2*b4 + b3 + 1) * q^23 + (b5 + b4 - b3) * q^24 + (4*b5 - 2*b4 - 3*b3 - 4*b2 - 3*b1 - 2) * q^25 + (-b5 + b4 + b3 + b2 + b1 + 1) * q^26 + (2*b5 - 3*b4 + 2*b3 - 2*b2 - 2*b1 - 1) * q^27 + (2*b5 + b3 - 4*b1 + 1) * q^29 + (3*b5 - 2*b4 - b3 - 2*b2 - b1 - 2) * q^30 + (b4 - 3*b3 - b2 + b1 - 6) * q^31 + q^32 + (b5 + 4*b3 - b2 - 4*b1 + 7) * q^33 + (-4*b5 - 2*b4 + 2*b3 + 2) * q^34 + (b5 - 2*b4 - 2*b1 - 1) * q^36 + (-3*b4 + 2*b3 + 2*b1 - 3) * q^37 + (b4 - b3 - b1 + 1) * q^38 + (-b5 + b4 + 2*b2 + 2*b1 + 1) * q^39 + (b5 - 2*b4 - b1) * q^40 + (4*b4 + b3 + b1 + 4) * q^41 + (-3*b5 - 6*b4 + b3 + b1 + 1) * q^43 + (-b5 - b3 + b2 - b1) * q^44 + (2*b5 - 4*b4 - 3*b3 - 3*b2 - b1 - 8) * q^45 + (-2*b5 + 2*b4 + b3 + 1) * q^46 + (b4 + b3 + 3*b2 + b1 + 1) * q^47 + (b5 + b4 - b3) * q^48 + (4*b5 - 2*b4 - 3*b3 - 4*b2 - 3*b1 - 2) * q^50 + (-2*b5 + 10*b4 + 4*b1 + 8) * q^51 + (-b5 + b4 + b3 + b2 + b1 + 1) * q^52 + (-3*b5 - b4 - b3 + 5*b1 - 1) * q^53 + (2*b5 - 3*b4 + 2*b3 - 2*b2 - 2*b1 - 1) * q^54 + (3*b4 - 2*b3 + 4*b2 + 3*b1 - 4) * q^55 + (b5 + b4 + 2*b3 - 3*b1 + 3) * q^57 + (2*b5 + b3 - 4*b1 + 1) * q^58 + (b4 - 6*b3 - 4*b2 + b1 - 1) * q^59 + (3*b5 - 2*b4 - b3 - 2*b2 - b1 - 2) * q^60 + (-3*b4 + 8*b3 + 2*b2 - 3*b1 + 5) * q^61 + (b4 - 3*b3 - b2 + b1 - 6) * q^62 + q^64 + (b4 + b3 + 3*b2 + b1 + 4) * q^65 + (b5 + 4*b3 - b2 - 4*b1 + 7) * q^66 + (-3*b4 + 2*b3 - 4*b2 - 3*b1) * q^67 + (-4*b5 - 2*b4 + 2*b3 + 2) * q^68 + (-4*b5 + 5*b4 + 3*b3 + 3*b2 + 2*b1 + 4) * q^69 + (-3*b4 + 2*b3 - 4*b2 - 3*b1 + 4) * q^71 + (b5 - 2*b4 - 2*b1 - 1) * q^72 + (2*b5 - 6*b4 - 3*b3 + 4*b1 - 3) * q^73 + (-3*b4 + 2*b3 + 2*b1 - 3) * q^74 + (3*b5 - 2*b4 - 2*b3 - 5*b2 - 5*b1 - 7) * q^75 + (b4 - b3 - b1 + 1) * q^76 + (-b5 + b4 + 2*b2 + 2*b1 + 1) * q^78 + (4*b4 - b3 + 7*b2 + 4*b1 - 4) * q^79 + (b5 - 2*b4 - b1) * q^80 + (-b4 - 2*b3 - b2 + 4*b1 - 10) * q^81 + (4*b4 + b3 + b1 + 4) * q^82 + (-b5 + b4 + 2*b3 - 3*b1 + 2) * q^83 + (-10*b5 + 8*b4 + 4*b3 + 10*b2 + 4*b1 + 8) * q^85 + (-3*b5 - 6*b4 + b3 + b1 + 1) * q^86 + (2*b5 - 3*b4 + 5*b3 + b2 - 2*b1 - 4) * q^87 + (-b5 - b3 + b2 - b1) * q^88 + (-3*b5 + 4*b4 + b3 + 3*b2 + b1 + 4) * q^89 + (2*b5 - 4*b4 - 3*b3 - 3*b2 - b1 - 8) * q^90 + (-2*b5 + 2*b4 + b3 + 1) * q^92 + (-5*b5 - 4*b4 + 4*b3 - 2*b2 - 4*b1 + 4) * q^93 + (b4 + b3 + 3*b2 + b1 + 1) * q^94 + (b4 + 2*b2 + b1 - 1) * q^95 + (b5 + b4 - b3) * q^96 + (2*b5 - 8*b4 - 2*b3 + 2*b1 - 2) * q^97 + (3*b5 + 7*b4 - b3 + 4*b2 + 5*b1 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 5 q^{5} + 2 q^{6} + 6 q^{8} - 4 q^{9}+O(q^{10})$$ 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 5 * q^5 + 2 * q^6 + 6 * q^8 - 4 * q^9 $$6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 5 q^{5} + 2 q^{6} + 6 q^{8} - 4 q^{9} + 5 q^{10} - q^{11} + 2 q^{12} + 2 q^{13} - 2 q^{15} + 6 q^{16} + 4 q^{17} - 4 q^{18} + 3 q^{19} + 5 q^{20} - q^{22} - 7 q^{23} + 2 q^{24} - 2 q^{25} + 2 q^{26} - 7 q^{27} - 5 q^{29} - 2 q^{30} - 28 q^{31} + 6 q^{32} + 19 q^{33} + 4 q^{34} - 4 q^{36} - 9 q^{37} + 3 q^{38} + 9 q^{39} + 5 q^{40} + 12 q^{41} + 18 q^{43} - q^{44} - 29 q^{45} - 7 q^{46} + 6 q^{47} + 2 q^{48} - 2 q^{50} + 26 q^{51} + 2 q^{52} + 9 q^{53} - 7 q^{54} - 14 q^{55} + 2 q^{57} - 5 q^{58} + 8 q^{59} - 2 q^{60} + 8 q^{61} - 28 q^{62} + 6 q^{64} + 24 q^{65} + 19 q^{66} - 10 q^{67} + 4 q^{68} + q^{69} + 14 q^{71} - 4 q^{72} + 25 q^{73} - 9 q^{74} - 44 q^{75} + 3 q^{76} + 9 q^{78} - 14 q^{79} + 5 q^{80} - 40 q^{81} + 12 q^{82} - 8 q^{83} + 14 q^{85} + 18 q^{86} - 31 q^{87} - q^{88} + 9 q^{89} - 29 q^{90} - 7 q^{92} + 6 q^{94} - 4 q^{95} + 2 q^{96} + 28 q^{97} - 41 q^{99}+O(q^{100})$$ 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 5 * q^5 + 2 * q^6 + 6 * q^8 - 4 * q^9 + 5 * q^10 - q^11 + 2 * q^12 + 2 * q^13 - 2 * q^15 + 6 * q^16 + 4 * q^17 - 4 * q^18 + 3 * q^19 + 5 * q^20 - q^22 - 7 * q^23 + 2 * q^24 - 2 * q^25 + 2 * q^26 - 7 * q^27 - 5 * q^29 - 2 * q^30 - 28 * q^31 + 6 * q^32 + 19 * q^33 + 4 * q^34 - 4 * q^36 - 9 * q^37 + 3 * q^38 + 9 * q^39 + 5 * q^40 + 12 * q^41 + 18 * q^43 - q^44 - 29 * q^45 - 7 * q^46 + 6 * q^47 + 2 * q^48 - 2 * q^50 + 26 * q^51 + 2 * q^52 + 9 * q^53 - 7 * q^54 - 14 * q^55 + 2 * q^57 - 5 * q^58 + 8 * q^59 - 2 * q^60 + 8 * q^61 - 28 * q^62 + 6 * q^64 + 24 * q^65 + 19 * q^66 - 10 * q^67 + 4 * q^68 + q^69 + 14 * q^71 - 4 * q^72 + 25 * q^73 - 9 * q^74 - 44 * q^75 + 3 * q^76 + 9 * q^78 - 14 * q^79 + 5 * q^80 - 40 * q^81 + 12 * q^82 - 8 * q^83 + 14 * q^85 + 18 * q^86 - 31 * q^87 - q^88 + 9 * q^89 - 29 * q^90 - 7 * q^92 + 6 * q^94 - 4 * q^95 + 2 * q^96 + 28 * q^97 - 41 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3$$ (v^5 - v^4 + 5*v^3 + v^2 + 6) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3$$ (-v^5 + v^4 - 5*v^3 + 2*v^2 - 3*v) / 3 $$\beta_{4}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3$$ (-2*v^5 + 5*v^4 - 16*v^3 + 19*v^2 - 21*v + 6) / 3 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3$$ (2*v^5 - 5*v^4 + 19*v^3 - 22*v^2 + 33*v - 9) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta _1 - 2$$ b3 + b2 + b1 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 - 1$$ b5 + b4 + b3 + b2 - 3*b1 - 1 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 6\beta _1 + 6$$ 2*b5 + 3*b4 - 5*b3 - 3*b2 - 6*b1 + 6 $$\nu^{5}$$ $$=$$ $$-3\beta_{5} - 2\beta_{4} - 11\beta_{3} - 6\beta_{2} + 8\beta _1 + 7$$ -3*b5 - 2*b4 - 11*b3 - 6*b2 + 8*b1 + 7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-1 - \beta_{4}$$ $$-1 - \beta_{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
373.1
 0.5 − 1.41036i 0.5 + 2.05195i 0.5 + 0.224437i 0.5 + 1.41036i 0.5 − 2.05195i 0.5 − 0.224437i
1.00000 −1.09097 1.34528i 1.00000 0.880438 + 1.52496i −1.09097 1.34528i 0 1.00000 −0.619562 + 2.93533i 0.880438 + 1.52496i
373.2 1.00000 0.796790 1.53790i 1.00000 −0.230252 0.398809i 0.796790 1.53790i 0 1.00000 −1.73025 2.45076i −0.230252 0.398809i
373.3 1.00000 1.29418 + 1.15113i 1.00000 1.84981 + 3.20397i 1.29418 + 1.15113i 0 1.00000 0.349814 + 2.97954i 1.84981 + 3.20397i
655.1 1.00000 −1.09097 + 1.34528i 1.00000 0.880438 1.52496i −1.09097 + 1.34528i 0 1.00000 −0.619562 2.93533i 0.880438 1.52496i
655.2 1.00000 0.796790 + 1.53790i 1.00000 −0.230252 + 0.398809i 0.796790 + 1.53790i 0 1.00000 −1.73025 + 2.45076i −0.230252 + 0.398809i
655.3 1.00000 1.29418 1.15113i 1.00000 1.84981 3.20397i 1.29418 1.15113i 0 1.00000 0.349814 2.97954i 1.84981 3.20397i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 373.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.e.p 6
3.b odd 2 1 2646.2.e.o 6
7.b odd 2 1 126.2.e.d 6
7.c even 3 1 882.2.f.m 6
7.c even 3 1 882.2.h.o 6
7.d odd 6 1 126.2.h.c yes 6
7.d odd 6 1 882.2.f.l 6
9.c even 3 1 882.2.h.o 6
9.d odd 6 1 2646.2.h.p 6
21.c even 2 1 378.2.e.c 6
21.g even 6 1 378.2.h.d 6
21.g even 6 1 2646.2.f.o 6
21.h odd 6 1 2646.2.f.n 6
21.h odd 6 1 2646.2.h.p 6
28.d even 2 1 1008.2.q.h 6
28.f even 6 1 1008.2.t.g 6
63.g even 3 1 882.2.f.m 6
63.h even 3 1 inner 882.2.e.p 6
63.h even 3 1 7938.2.a.by 3
63.i even 6 1 378.2.e.c 6
63.i even 6 1 7938.2.a.bu 3
63.j odd 6 1 2646.2.e.o 6
63.j odd 6 1 7938.2.a.bx 3
63.k odd 6 1 882.2.f.l 6
63.k odd 6 1 1134.2.g.k 6
63.l odd 6 1 126.2.h.c yes 6
63.l odd 6 1 1134.2.g.k 6
63.n odd 6 1 2646.2.f.n 6
63.o even 6 1 378.2.h.d 6
63.o even 6 1 1134.2.g.n 6
63.s even 6 1 1134.2.g.n 6
63.s even 6 1 2646.2.f.o 6
63.t odd 6 1 126.2.e.d 6
63.t odd 6 1 7938.2.a.cb 3
84.h odd 2 1 3024.2.q.h 6
84.j odd 6 1 3024.2.t.g 6
252.r odd 6 1 3024.2.q.h 6
252.s odd 6 1 3024.2.t.g 6
252.bi even 6 1 1008.2.t.g 6
252.bj even 6 1 1008.2.q.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.d 6 7.b odd 2 1
126.2.e.d 6 63.t odd 6 1
126.2.h.c yes 6 7.d odd 6 1
126.2.h.c yes 6 63.l odd 6 1
378.2.e.c 6 21.c even 2 1
378.2.e.c 6 63.i even 6 1
378.2.h.d 6 21.g even 6 1
378.2.h.d 6 63.o even 6 1
882.2.e.p 6 1.a even 1 1 trivial
882.2.e.p 6 63.h even 3 1 inner
882.2.f.l 6 7.d odd 6 1
882.2.f.l 6 63.k odd 6 1
882.2.f.m 6 7.c even 3 1
882.2.f.m 6 63.g even 3 1
882.2.h.o 6 7.c even 3 1
882.2.h.o 6 9.c even 3 1
1008.2.q.h 6 28.d even 2 1
1008.2.q.h 6 252.bj even 6 1
1008.2.t.g 6 28.f even 6 1
1008.2.t.g 6 252.bi even 6 1
1134.2.g.k 6 63.k odd 6 1
1134.2.g.k 6 63.l odd 6 1
1134.2.g.n 6 63.o even 6 1
1134.2.g.n 6 63.s even 6 1
2646.2.e.o 6 3.b odd 2 1
2646.2.e.o 6 63.j odd 6 1
2646.2.f.n 6 21.h odd 6 1
2646.2.f.n 6 63.n odd 6 1
2646.2.f.o 6 21.g even 6 1
2646.2.f.o 6 63.s even 6 1
2646.2.h.p 6 9.d odd 6 1
2646.2.h.p 6 21.h odd 6 1
3024.2.q.h 6 84.h odd 2 1
3024.2.q.h 6 252.r odd 6 1
3024.2.t.g 6 84.j odd 6 1
3024.2.t.g 6 252.s odd 6 1
7938.2.a.bu 3 63.i even 6 1
7938.2.a.bx 3 63.j odd 6 1
7938.2.a.by 3 63.h even 3 1
7938.2.a.cb 3 63.t odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{6} - 5T_{5}^{5} + 21T_{5}^{4} - 26T_{5}^{3} + 31T_{5}^{2} + 12T_{5} + 9$$ T5^6 - 5*T5^5 + 21*T5^4 - 26*T5^3 + 31*T5^2 + 12*T5 + 9 $$T_{11}^{6} + T_{11}^{5} + 27T_{11}^{4} - 92T_{11}^{3} + 643T_{11}^{2} - 858T_{11} + 1089$$ T11^6 + T11^5 + 27*T11^4 - 92*T11^3 + 643*T11^2 - 858*T11 + 1089 $$T_{13}^{6} - 2T_{13}^{5} + 7T_{13}^{4} + 15T_{13}^{2} - 9T_{13} + 9$$ T13^6 - 2*T13^5 + 7*T13^4 + 15*T13^2 - 9*T13 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{6}$$
$3$ $$T^{6} - 2 T^{5} + \cdots + 27$$
$5$ $$T^{6} - 5 T^{5} + \cdots + 9$$
$7$ $$T^{6}$$
$11$ $$T^{6} + T^{5} + \cdots + 1089$$
$13$ $$T^{6} - 2 T^{5} + \cdots + 9$$
$17$ $$T^{6} - 4 T^{5} + \cdots + 28224$$
$19$ $$T^{6} - 3 T^{5} + \cdots + 49$$
$23$ $$T^{6} + 7 T^{5} + \cdots + 9$$
$29$ $$T^{6} + 5 T^{5} + \cdots + 1089$$
$31$ $$(T^{3} + 14 T^{2} + \cdots - 27)^{2}$$
$37$ $$T^{6} + 9 T^{5} + \cdots + 5329$$
$41$ $$T^{6} - 12 T^{5} + \cdots + 729$$
$43$ $$T^{6} - 18 T^{5} + \cdots + 1$$
$47$ $$(T^{3} - 3 T^{2} - 24 T - 27)^{2}$$
$53$ $$T^{6} - 9 T^{5} + \cdots + 81$$
$59$ $$(T^{3} - 4 T^{2} + \cdots - 177)^{2}$$
$61$ $$(T^{3} - 4 T^{2} + \cdots + 717)^{2}$$
$67$ $$(T^{3} + 5 T^{2} + \cdots - 149)^{2}$$
$71$ $$(T^{3} - 7 T^{2} - 50 T + 99)^{2}$$
$73$ $$T^{6} - 25 T^{5} + \cdots + 2401$$
$79$ $$(T^{3} + 7 T^{2} + \cdots - 771)^{2}$$
$83$ $$T^{6} + 8 T^{5} + \cdots + 8649$$
$89$ $$T^{6} - 9 T^{5} + \cdots + 3969$$
$97$ $$T^{6} - 28 T^{5} + \cdots + 287296$$