# Properties

 Label 1323.2.g.c Level $1323$ Weight $2$ Character orbit 1323.g Analytic conductor $10.564$ 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] = [1323,2,Mod(361,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 4]))

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

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

chi := DirichletCharacter("1323.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.g (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: $$10.5642081874$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{5} + \beta_{4} + \beta_{2} - 1) q^{4} + (\beta_{3} + 2) q^{5} + (\beta_{3} - \beta_1 + 2) q^{8}+O(q^{10})$$ q + (-b5 + b1) * q^2 + (b5 + b4 + b2 - 1) * q^4 + (b3 + 2) * q^5 + (b3 - b1 + 2) * q^8 $$q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{5} + \beta_{4} + \beta_{2} - 1) q^{4} + (\beta_{3} + 2) q^{5} + (\beta_{3} - \beta_1 + 2) q^{8} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_1) q^{10} + ( - \beta_{3} + 2 \beta_1 + 1) q^{11} - \beta_{4} q^{13} + ( - 2 \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{16} + (\beta_{5} - 4 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{17} + ( - \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{19} + (2 \beta_{5} + 5 \beta_{4} + \beta_{2} - 5) q^{20} + (2 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{22} + ( - 2 \beta_{3} - \beta_1 - 1) q^{23} + (2 \beta_{3} + \beta_1 + 3) q^{25} + \beta_{5} q^{26} - \beta_{5} q^{29} + (2 \beta_{5} - 2 \beta_{4} - \beta_{2} + 2) q^{31} + ( - \beta_{5} + 3 \beta_{4} - 3) q^{32} + (2 \beta_{5} - 2 \beta_{4} - \beta_{2} + 2) q^{34} + 3 \beta_{2} q^{37} + ( - \beta_{3} + 3 \beta_1 - 2) q^{38} + (2 \beta_{3} - 2 \beta_1 + 7) q^{40} + ( - 7 \beta_{4} + \beta_{3} + \beta_{2}) q^{41} + ( - 4 \beta_{5} - \beta_{2}) q^{43} + ( - 5 \beta_{5} - 6 \beta_{4} + 6) q^{44} + (2 \beta_{5} - 5 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1) q^{46} + ( - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{47} + ( - 4 \beta_{5} + 5 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_1) q^{50} + (\beta_{3} - \beta_1 + 1) q^{52} + ( - \beta_{5} - 5 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{53} + (\beta_{3} + 5 \beta_1) q^{55} + ( - \beta_{3} + \beta_1 - 3) q^{58} + ( - \beta_{5} + 4 \beta_{4} + 2 \beta_{2} - 4) q^{59} + (\beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{61} + (2 \beta_{3} + \beta_1 + 7) q^{62} + (\beta_{3} + 2 \beta_1 - 3) q^{64} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2}) q^{65} + ( - 7 \beta_{5} + 2 \beta_{4} - \beta_{2} - 2) q^{67} + (4 \beta_{3} - \beta_1 - 1) q^{68} + (2 \beta_{3} + \beta_1 - 2) q^{71} + ( - \beta_{5} + \beta_{4} + 5 \beta_{3} + 5 \beta_{2} + \beta_1) q^{73} + ( - 3 \beta_1 - 3) q^{74} + (4 \beta_{5} + 6 \beta_{4} + \beta_{3} + \beta_{2} - 4 \beta_1) q^{76} + ( - 5 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_1) q^{79} + ( - 7 \beta_{5} + 6 \beta_{4} + 7 \beta_1) q^{80} + (6 \beta_{5} + \beta_{4} - 1) q^{82} + ( - \beta_{5} + 4 \beta_{4} - \beta_{2} - 4) q^{83} + (2 \beta_{5} - 5 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{85} + ( - 4 \beta_{3} + 5 \beta_1 - 11) q^{86} + ( - \beta_{3} + 7 \beta_1 - 5) q^{88} + (6 \beta_{5} - \beta_{4} + \beta_{2} + 1) q^{89} + (2 \beta_{5} - 5 \beta_{4} + 2 \beta_{2} + 5) q^{92} + (9 \beta_{5} + 6 \beta_{4} + 3 \beta_{2} - 6) q^{94} + ( - 2 \beta_{5} - 5 \beta_{4} - \beta_{2} + 5) q^{95} + ( - 5 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 2) q^{97}+O(q^{100})$$ q + (-b5 + b1) * q^2 + (b5 + b4 + b2 - 1) * q^4 + (b3 + 2) * q^5 + (b3 - b1 + 2) * q^8 + (-3*b5 + b4 + 3*b1) * q^10 + (-b3 + 2*b1 + 1) * q^11 - b4 * q^13 + (-2*b5 + b3 + b2 + 2*b1) * q^16 + (b5 - 4*b4 + b3 + b2 - b1) * q^17 + (-b5 - b4 - b2 + 1) * q^19 + (2*b5 + 5*b4 + b2 - 5) * q^20 + (2*b5 + 5*b4 + 2*b3 + 2*b2 - 2*b1) * q^22 + (-2*b3 - b1 - 1) * q^23 + (2*b3 + b1 + 3) * q^25 + b5 * q^26 - b5 * q^29 + (2*b5 - 2*b4 - b2 + 2) * q^31 + (-b5 + 3*b4 - 3) * q^32 + (2*b5 - 2*b4 - b2 + 2) * q^34 + 3*b2 * q^37 + (-b3 + 3*b1 - 2) * q^38 + (2*b3 - 2*b1 + 7) * q^40 + (-7*b4 + b3 + b2) * q^41 + (-4*b5 - b2) * q^43 + (-5*b5 - 6*b4 + 6) * q^44 + (2*b5 - 5*b4 - b3 - b2 - 2*b1) * q^46 + (-3*b5 - 3*b4 - 3*b3 - 3*b2 + 3*b1) * q^47 + (-4*b5 + 5*b4 + b3 + b2 + 4*b1) * q^50 + (b3 - b1 + 1) * q^52 + (-b5 - 5*b4 + 2*b3 + 2*b2 + b1) * q^53 + (b3 + 5*b1) * q^55 + (-b3 + b1 - 3) * q^58 + (-b5 + 4*b4 + 2*b2 - 4) * q^59 + (b5 + b4 - 2*b3 - 2*b2 - b1) * q^61 + (2*b3 + b1 + 7) * q^62 + (b3 + 2*b1 - 3) * q^64 + (-2*b4 - b3 - b2) * q^65 + (-7*b5 + 2*b4 - b2 - 2) * q^67 + (4*b3 - b1 - 1) * q^68 + (2*b3 + b1 - 2) * q^71 + (-b5 + b4 + 5*b3 + 5*b2 + b1) * q^73 + (-3*b1 - 3) * q^74 + (4*b5 + 6*b4 + b3 + b2 - 4*b1) * q^76 + (-5*b5 - 3*b4 + b3 + b2 + 5*b1) * q^79 + (-7*b5 + 6*b4 + 7*b1) * q^80 + (6*b5 + b4 - 1) * q^82 + (-b5 + 4*b4 - b2 - 4) * q^83 + (2*b5 - 5*b4 - 4*b3 - 4*b2 - 2*b1) * q^85 + (-4*b3 + 5*b1 - 11) * q^86 + (-b3 + 7*b1 - 5) * q^88 + (6*b5 - b4 + b2 + 1) * q^89 + (2*b5 - 5*b4 + 2*b2 + 5) * q^92 + (9*b5 + 6*b4 + 3*b2 - 6) * q^94 + (-2*b5 - 5*b4 - b2 + 5) * q^95 + (-5*b5 + 2*b4 - 2*b2 - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{2} - 3 q^{4} + 10 q^{5} + 12 q^{8}+O(q^{10})$$ 6 * q - q^2 - 3 * q^4 + 10 * q^5 + 12 * q^8 $$6 q - q^{2} - 3 q^{4} + 10 q^{5} + 12 q^{8} + 4 q^{11} - 3 q^{13} - 3 q^{16} - 12 q^{17} + 3 q^{19} - 16 q^{20} + 15 q^{22} + 12 q^{25} - q^{26} + q^{29} + 3 q^{31} - 8 q^{32} + 3 q^{34} + 3 q^{37} - 16 q^{38} + 42 q^{40} - 22 q^{41} + 3 q^{43} + 23 q^{44} - 12 q^{46} - 9 q^{47} + 10 q^{50} + 6 q^{52} - 18 q^{53} - 12 q^{55} - 18 q^{58} - 9 q^{59} + 6 q^{61} + 36 q^{62} - 24 q^{64} - 5 q^{65} - 12 q^{68} - 18 q^{71} - 3 q^{73} - 12 q^{74} + 21 q^{76} - 15 q^{79} + 11 q^{80} - 9 q^{82} - 12 q^{83} - 9 q^{85} - 68 q^{86} - 42 q^{88} - 2 q^{89} + 15 q^{92} - 24 q^{94} + 16 q^{95} - 3 q^{97}+O(q^{100})$$ 6 * q - q^2 - 3 * q^4 + 10 * q^5 + 12 * q^8 + 4 * q^11 - 3 * q^13 - 3 * q^16 - 12 * q^17 + 3 * q^19 - 16 * q^20 + 15 * q^22 + 12 * q^25 - q^26 + q^29 + 3 * q^31 - 8 * q^32 + 3 * q^34 + 3 * q^37 - 16 * q^38 + 42 * q^40 - 22 * q^41 + 3 * q^43 + 23 * q^44 - 12 * q^46 - 9 * q^47 + 10 * q^50 + 6 * q^52 - 18 * q^53 - 12 * q^55 - 18 * q^58 - 9 * q^59 + 6 * q^61 + 36 * q^62 - 24 * q^64 - 5 * q^65 - 12 * q^68 - 18 * q^71 - 3 * q^73 - 12 * q^74 + 21 * q^76 - 15 * q^79 + 11 * q^80 - 9 * q^82 - 12 * q^83 - 9 * q^85 - 68 * q^86 - 42 * q^88 - 2 * q^89 + 15 * q^92 - 24 * q^94 + 16 * q^95 - 3 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 2$$ v^2 - v + 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3$$ (-v^5 + v^4 - 8*v^3 + 5*v^2 - 18*v + 6) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3$$ v^4 - 2*v^3 + 6*v^2 - 5*v + 3 $$\beta_{4}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3$$ (-2*v^5 + 5*v^4 - 16*v^3 + 19*v^2 - 21*v + 9) / 3 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3$$ (2*v^5 - 5*v^4 + 19*v^3 - 22*v^2 + 30*v - 9) / 3
 $$\nu$$ $$=$$ $$( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3$$ (-2*b5 - b4 - b3 - 2*b2 + b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3$$ (-2*b5 - b4 - b3 - 2*b2 + 4*b1 - 4) / 3 $$\nu^{3}$$ $$=$$ $$( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3$$ (7*b5 + 5*b4 + 2*b3 + 4*b2 + b1 - 10) / 3 $$\nu^{4}$$ $$=$$ $$( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3$$ (16*b5 + 11*b4 + 8*b3 + 10*b2 - 17*b1 + 5) / 3 $$\nu^{5}$$ $$=$$ $$( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3$$ (-14*b5 - 16*b4 + 5*b3 - 5*b2 - 23*b1 + 47) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-\beta_{4}$$ $$-1 + \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 2.05195i 0.5 − 1.41036i 0.5 + 0.224437i 0.5 − 2.05195i 0.5 + 1.41036i 0.5 − 0.224437i
−1.23025 + 2.13086i 0 −2.02704 3.51094i 2.59358 0 0 5.05408 0 −3.19076 + 5.52655i
361.2 −0.119562 + 0.207087i 0 0.971410 + 1.68253i −1.18194 0 0 −0.942820 0 0.141315 0.244765i
361.3 0.849814 1.47192i 0 −0.444368 0.769668i 3.58836 0 0 1.88874 0 3.04944 5.28179i
667.1 −1.23025 2.13086i 0 −2.02704 + 3.51094i 2.59358 0 0 5.05408 0 −3.19076 5.52655i
667.2 −0.119562 0.207087i 0 0.971410 1.68253i −1.18194 0 0 −0.942820 0 0.141315 + 0.244765i
667.3 0.849814 + 1.47192i 0 −0.444368 + 0.769668i 3.58836 0 0 1.88874 0 3.04944 + 5.28179i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.3 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.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.g.c 6
3.b odd 2 1 441.2.g.e 6
7.b odd 2 1 1323.2.g.b 6
7.c even 3 1 189.2.f.a 6
7.c even 3 1 1323.2.h.d 6
7.d odd 6 1 1323.2.f.c 6
7.d odd 6 1 1323.2.h.e 6
9.c even 3 1 1323.2.h.d 6
9.d odd 6 1 441.2.h.c 6
21.c even 2 1 441.2.g.d 6
21.g even 6 1 441.2.f.d 6
21.g even 6 1 441.2.h.b 6
21.h odd 6 1 63.2.f.b 6
21.h odd 6 1 441.2.h.c 6
28.g odd 6 1 3024.2.r.g 6
63.g even 3 1 567.2.a.g 3
63.g even 3 1 inner 1323.2.g.c 6
63.h even 3 1 189.2.f.a 6
63.i even 6 1 441.2.f.d 6
63.j odd 6 1 63.2.f.b 6
63.k odd 6 1 1323.2.g.b 6
63.k odd 6 1 3969.2.a.p 3
63.l odd 6 1 1323.2.h.e 6
63.n odd 6 1 441.2.g.e 6
63.n odd 6 1 567.2.a.d 3
63.o even 6 1 441.2.h.b 6
63.s even 6 1 441.2.g.d 6
63.s even 6 1 3969.2.a.m 3
63.t odd 6 1 1323.2.f.c 6
84.n even 6 1 1008.2.r.k 6
252.o even 6 1 9072.2.a.bq 3
252.u odd 6 1 3024.2.r.g 6
252.bb even 6 1 1008.2.r.k 6
252.bl odd 6 1 9072.2.a.cd 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 21.h odd 6 1
63.2.f.b 6 63.j odd 6 1
189.2.f.a 6 7.c even 3 1
189.2.f.a 6 63.h even 3 1
441.2.f.d 6 21.g even 6 1
441.2.f.d 6 63.i even 6 1
441.2.g.d 6 21.c even 2 1
441.2.g.d 6 63.s even 6 1
441.2.g.e 6 3.b odd 2 1
441.2.g.e 6 63.n odd 6 1
441.2.h.b 6 21.g even 6 1
441.2.h.b 6 63.o even 6 1
441.2.h.c 6 9.d odd 6 1
441.2.h.c 6 21.h odd 6 1
567.2.a.d 3 63.n odd 6 1
567.2.a.g 3 63.g even 3 1
1008.2.r.k 6 84.n even 6 1
1008.2.r.k 6 252.bb even 6 1
1323.2.f.c 6 7.d odd 6 1
1323.2.f.c 6 63.t odd 6 1
1323.2.g.b 6 7.b odd 2 1
1323.2.g.b 6 63.k odd 6 1
1323.2.g.c 6 1.a even 1 1 trivial
1323.2.g.c 6 63.g even 3 1 inner
1323.2.h.d 6 7.c even 3 1
1323.2.h.d 6 9.c even 3 1
1323.2.h.e 6 7.d odd 6 1
1323.2.h.e 6 63.l odd 6 1
3024.2.r.g 6 28.g odd 6 1
3024.2.r.g 6 252.u odd 6 1
3969.2.a.m 3 63.s even 6 1
3969.2.a.p 3 63.k odd 6 1
9072.2.a.bq 3 252.o even 6 1
9072.2.a.cd 3 252.bl odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1323, [\chi])$$:

 $$T_{2}^{6} + T_{2}^{5} + 5T_{2}^{4} - 2T_{2}^{3} + 17T_{2}^{2} + 4T_{2} + 1$$ T2^6 + T2^5 + 5*T2^4 - 2*T2^3 + 17*T2^2 + 4*T2 + 1 $$T_{5}^{3} - 5T_{5}^{2} + 2T_{5} + 11$$ T5^3 - 5*T5^2 + 2*T5 + 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{5} + 5 T^{4} - 2 T^{3} + 17 T^{2} + \cdots + 1$$
$3$ $$T^{6}$$
$5$ $$(T^{3} - 5 T^{2} + 2 T + 11)^{2}$$
$7$ $$T^{6}$$
$11$ $$(T^{3} - 2 T^{2} - 19 T + 47)^{2}$$
$13$ $$(T^{2} + T + 1)^{3}$$
$17$ $$T^{6} + 12 T^{5} + 105 T^{4} + \cdots + 729$$
$19$ $$T^{6} - 3 T^{5} + 15 T^{4} + 4 T^{3} + \cdots + 49$$
$23$ $$(T^{3} - 33 T + 9)^{2}$$
$29$ $$T^{6} - T^{5} + 5 T^{4} + 2 T^{3} + 17 T^{2} + \cdots + 1$$
$31$ $$T^{6} - 3 T^{5} + 33 T^{4} + 126 T^{3} + \cdots + 729$$
$37$ $$T^{6} - 3 T^{5} + 63 T^{4} + \cdots + 6561$$
$41$ $$T^{6} + 22 T^{5} + 329 T^{4} + \cdots + 124609$$
$43$ $$T^{6} - 3 T^{5} + 75 T^{4} + \cdots + 14641$$
$47$ $$T^{6} + 9 T^{5} + 135 T^{4} + \cdots + 35721$$
$53$ $$T^{6} + 18 T^{5} + 249 T^{4} + \cdots + 81$$
$59$ $$T^{6} + 9 T^{5} + 87 T^{4} + \cdots + 3969$$
$61$ $$T^{6} - 6 T^{5} + 57 T^{4} + \cdots + 4489$$
$67$ $$T^{6} + 207 T^{4} + 1366 T^{3} + \cdots + 466489$$
$71$ $$(T^{3} + 9 T^{2} - 6 T - 81)^{2}$$
$73$ $$T^{6} + 3 T^{5} + 177 T^{4} + \cdots + 59049$$
$79$ $$T^{6} + 15 T^{5} + 273 T^{4} + \cdots + 591361$$
$83$ $$T^{6} + 12 T^{5} + 105 T^{4} + \cdots + 729$$
$89$ $$T^{6} + 2 T^{5} + 155 T^{4} + \cdots + 143641$$
$97$ $$T^{6} + 3 T^{5} + 123 T^{4} + \cdots + 363609$$