# Properties

 Label 585.2.j.f Level $585$ Weight $2$ Character orbit 585.j Analytic conductor $4.671$ 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] = [585,2,Mod(406,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("585.406");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.j (of order $$3$$, degree $$2$$, 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: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.1714608.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7$$ x^6 - 3*x^5 + 12*x^4 - 19*x^3 + 30*x^2 - 21*x + 7 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{5} + \beta_{4} - \beta_{2} - 2 \beta_1) q^{4} - q^{5} + ( - \beta_{5} + \beta_{4}) q^{7} + ( - 2 \beta_{3} - 2) q^{8}+O(q^{10})$$ q + (-b5 + b3) * q^2 + (-b5 + b4 - b2 - 2*b1) * q^4 - q^5 + (-b5 + b4) * q^7 + (-2*b3 - 2) * q^8 $$q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{5} + \beta_{4} - \beta_{2} - 2 \beta_1) q^{4} - q^{5} + ( - \beta_{5} + \beta_{4}) q^{7} + ( - 2 \beta_{3} - 2) q^{8} + (\beta_{5} - \beta_{3}) q^{10} + (2 \beta_{5} - 2 \beta_{3}) q^{11} + ( - 3 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{13} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 5) q^{14} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 4) q^{16} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 4 \beta_1) q^{17} + ( - \beta_{5} + 3 \beta_{4} - \beta_{2} - 2 \beta_1) q^{19} + (\beta_{5} - \beta_{4} + \beta_{2} + 2 \beta_1) q^{20} + (2 \beta_{5} - 6 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{22} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{23} + q^{25} + (5 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 7) q^{26} + (5 \beta_{5} - 4 \beta_{4} - 5 \beta_{3} - \beta_{2} + \beta_1 - 5) q^{28} + (2 \beta_{4} - \beta_{2} + \beta_1 + 1) q^{29} + (3 \beta_{4} + \beta_{3} + 6 \beta_{2} + 3 \beta_1 + 4) q^{31} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{32} + (\beta_{4} - 3 \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{34} + (\beta_{5} - \beta_{4}) q^{35} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{37} + ( - 6 \beta_{3} - 2) q^{38} + (2 \beta_{3} + 2) q^{40} + (2 \beta_{5} - 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{41} + (2 \beta_{4} - \beta_{2} - 2 \beta_1) q^{43} + (8 \beta_{3} + 4) q^{44} + ( - 4 \beta_{5} + 4 \beta_{4}) q^{46} + ( - 3 \beta_{3} + 4) q^{47} + (3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{49} + ( - \beta_{5} + \beta_{3}) q^{50} + ( - 7 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 5) q^{52} + 2 \beta_{3} q^{53} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{55} + (6 \beta_{5} - 8 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{56} + ( - \beta_{5} + \beta_{4} - \beta_{2} - 2 \beta_1) q^{58} + ( - 2 \beta_{5} + \beta_{4} - \beta_{2} - 2 \beta_1) q^{59} + (\beta_{5} - 2 \beta_{4} - \beta_{2} - 2 \beta_1) q^{61} + ( - 5 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{62} + (4 \beta_{3} - 4) q^{64} + (3 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{65} + ( - 3 \beta_{4} - 3 \beta_{2} + 3 \beta_1 - 6) q^{67} + ( - 14 \beta_{4} - 14) q^{68} + (\beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 5) q^{70} + ( - 2 \beta_{5} + \beta_{4} - 3 \beta_{2} - 6 \beta_1) q^{71} + ( - 3 \beta_{3} + 7) q^{73} + ( - 2 \beta_{5} + 4 \beta_{4}) q^{74} + (6 \beta_{5} - 16 \beta_{4} - 6 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 20) q^{76} + (2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 10) q^{77} + ( - 3 \beta_{4} - \beta_{3} - 6 \beta_{2} - 3 \beta_1 - 4) q^{79} + ( - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 4) q^{80} + (3 \beta_{5} - 5 \beta_{4} + \beta_{2} + 2 \beta_1) q^{82} + (2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 8) q^{83} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{85} + (\beta_{4} - 4 \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{86} + ( - 8 \beta_{5} + 16 \beta_{4} + 8 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 20) q^{88} + (10 \beta_{4} - \beta_{2} + \beta_1 + 9) q^{89} + ( - 3 \beta_{5} + 6 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{91} + ( - 4 \beta_{3} - 16) q^{92} + ( - \beta_{5} - 12 \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 15) q^{94} + (\beta_{5} - 3 \beta_{4} + \beta_{2} + 2 \beta_1) q^{95} + (4 \beta_{5} + 6 \beta_{4} + \beta_{2} + 2 \beta_1) q^{97} + (2 \beta_{5} - 8 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{98}+O(q^{100})$$ q + (-b5 + b3) * q^2 + (-b5 + b4 - b2 - 2*b1) * q^4 - q^5 + (-b5 + b4) * q^7 + (-2*b3 - 2) * q^8 + (b5 - b3) * q^10 + (2*b5 - 2*b3) * q^11 + (-3*b4 + b3 - b2 - b1 - 1) * q^13 + (-b4 - 2*b3 - 2*b2 - b1 - 5) * q^14 + (2*b5 - 4*b4 - 2*b3 - 4) * q^16 + (-b5 - 2*b4 - 2*b2 - 4*b1) * q^17 + (-b5 + 3*b4 - b2 - 2*b1) * q^19 + (b5 - b4 + b2 + 2*b1) * q^20 + (2*b5 - 6*b4 + 2*b2 + 4*b1) * q^22 + (-2*b5 + 2*b3 + 2*b2 - 2*b1 + 2) * q^23 + q^25 + (5*b4 + 2*b3 + 2*b2 - b1 + 7) * q^26 + (5*b5 - 4*b4 - 5*b3 - b2 + b1 - 5) * q^28 + (2*b4 - b2 + b1 + 1) * q^29 + (3*b4 + b3 + 6*b2 + 3*b1 + 4) * q^31 + (2*b5 - 2*b4 + 2*b2 + 4*b1) * q^32 + (b4 - 3*b3 + 2*b2 + b1 + 1) * q^34 + (b5 - b4) * q^35 + (-2*b5 - 2*b4 + 2*b3 + 2*b2 - 2*b1) * q^37 + (-6*b3 - 2) * q^38 + (2*b3 + 2) * q^40 + (2*b5 - 2*b3 - b2 + b1 - 1) * q^41 + (2*b4 - b2 - 2*b1) * q^43 + (8*b3 + 4) * q^44 + (-4*b5 + 4*b4) * q^46 + (-3*b3 + 4) * q^47 + (3*b5 + 2*b4 - 3*b3 - b2 + b1 + 1) * q^49 + (-b5 + b3) * q^50 + (-7*b5 + 2*b4 + 3*b3 + 3*b2 - b1 + 5) * q^52 + 2*b3 * q^53 + (-2*b5 + 2*b3) * q^55 + (6*b5 - 8*b4 + 2*b2 + 4*b1) * q^56 + (-b5 + b4 - b2 - 2*b1) * q^58 + (-2*b5 + b4 - b2 - 2*b1) * q^59 + (b5 - 2*b4 - b2 - 2*b1) * q^61 + (-5*b5 - 2*b4 + 5*b3 - 2*b2 + 2*b1 - 4) * q^62 + (4*b3 - 4) * q^64 + (3*b4 - b3 + b2 + b1 + 1) * q^65 + (-3*b4 - 3*b2 + 3*b1 - 6) * q^67 + (-14*b4 - 14) * q^68 + (b4 + 2*b3 + 2*b2 + b1 + 5) * q^70 + (-2*b5 + b4 - 3*b2 - 6*b1) * q^71 + (-3*b3 + 7) * q^73 + (-2*b5 + 4*b4) * q^74 + (6*b5 - 16*b4 - 6*b3 - 4*b2 + 4*b1 - 20) * q^76 + (2*b4 + 4*b3 + 4*b2 + 2*b1 + 10) * q^77 + (-3*b4 - b3 - 6*b2 - 3*b1 - 4) * q^79 + (-2*b5 + 4*b4 + 2*b3 + 4) * q^80 + (3*b5 - 5*b4 + b2 + 2*b1) * q^82 + (2*b4 + 2*b3 + 4*b2 + 2*b1 + 8) * q^83 + (b5 + 2*b4 + 2*b2 + 4*b1) * q^85 + (b4 - 4*b3 + 2*b2 + b1 + 3) * q^86 + (-8*b5 + 16*b4 + 8*b3 + 4*b2 - 4*b1 + 20) * q^88 + (10*b4 - b2 + b1 + 9) * q^89 + (-3*b5 + 6*b4 + 2*b3 - b2 - 2*b1 + 3) * q^91 + (-4*b3 - 16) * q^92 + (-b5 - 12*b4 + b3 - 3*b2 + 3*b1 - 15) * q^94 + (b5 - 3*b4 + b2 + 2*b1) * q^95 + (4*b5 + 6*b4 + b2 + 2*b1) * q^97 + (2*b5 - 8*b4 + 2*b2 + 4*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} - 6 q^{5} - 3 q^{7} - 12 q^{8}+O(q^{10})$$ 6 * q - 6 * q^4 - 6 * q^5 - 3 * q^7 - 12 * q^8 $$6 q - 6 q^{4} - 6 q^{5} - 3 q^{7} - 12 q^{8} + 3 q^{13} - 24 q^{14} - 12 q^{16} - 12 q^{19} + 6 q^{20} + 24 q^{22} + 6 q^{25} + 18 q^{26} - 12 q^{28} + 6 q^{29} + 6 q^{31} + 12 q^{32} + 3 q^{35} - 6 q^{37} - 12 q^{38} + 12 q^{40} - 9 q^{43} + 24 q^{44} - 12 q^{46} + 24 q^{47} + 6 q^{49} + 12 q^{52} + 30 q^{56} - 6 q^{58} - 6 q^{59} + 3 q^{61} - 6 q^{62} - 24 q^{64} - 3 q^{65} - 9 q^{67} - 42 q^{68} + 24 q^{70} - 12 q^{71} + 42 q^{73} - 12 q^{74} - 48 q^{76} + 48 q^{77} - 6 q^{79} + 12 q^{80} + 18 q^{82} + 36 q^{83} + 12 q^{86} + 48 q^{88} + 30 q^{89} - 3 q^{91} - 96 q^{92} - 36 q^{94} + 12 q^{95} - 15 q^{97} + 30 q^{98}+O(q^{100})$$ 6 * q - 6 * q^4 - 6 * q^5 - 3 * q^7 - 12 * q^8 + 3 * q^13 - 24 * q^14 - 12 * q^16 - 12 * q^19 + 6 * q^20 + 24 * q^22 + 6 * q^25 + 18 * q^26 - 12 * q^28 + 6 * q^29 + 6 * q^31 + 12 * q^32 + 3 * q^35 - 6 * q^37 - 12 * q^38 + 12 * q^40 - 9 * q^43 + 24 * q^44 - 12 * q^46 + 24 * q^47 + 6 * q^49 + 12 * q^52 + 30 * q^56 - 6 * q^58 - 6 * q^59 + 3 * q^61 - 6 * q^62 - 24 * q^64 - 3 * q^65 - 9 * q^67 - 42 * q^68 + 24 * q^70 - 12 * q^71 + 42 * q^73 - 12 * q^74 - 48 * q^76 + 48 * q^77 - 6 * q^79 + 12 * q^80 + 18 * q^82 + 36 * q^83 + 12 * q^86 + 48 * q^88 + 30 * q^89 - 3 * q^91 - 96 * q^92 - 36 * q^94 + 12 * q^95 - 15 * q^97 + 30 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} + \nu^{4} + 2\nu^{3} + 10\nu^{2} - 7\nu ) / 7$$ (v^5 + v^4 + 2*v^3 + 10*v^2 - 7*v) / 7 $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu + 3$$ v^2 - v + 3 $$\beta_{4}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 18\nu^{3} + 22\nu^{2} - 28\nu + 7 ) / 7$$ (-2*v^5 + 5*v^4 - 18*v^3 + 22*v^2 - 28*v + 7) / 7 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 25\nu^{3} - 29\nu^{2} + 56\nu - 14 ) / 7$$ (2*v^5 - 5*v^4 + 25*v^3 - 29*v^2 + 56*v - 14) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta _1 - 3$$ b3 + b1 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} - 3\beta _1 - 2$$ b5 + b4 + b3 - 3*b1 - 2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 3\beta_{4} - 4\beta_{3} + 2\beta_{2} - 6\beta _1 + 13$$ 2*b5 + 3*b4 - 4*b3 + 2*b2 - 6*b1 + 13 $$\nu^{5}$$ $$=$$ $$-4\beta_{5} - 5\beta_{4} - 8\beta_{3} + 5\beta_{2} + 9\beta _1 + 21$$ -4*b5 - 5*b4 - 8*b3 + 5*b2 + 9*b1 + 21

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$496$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
406.1
 0.5 − 2.23871i 0.5 + 1.75780i 0.5 − 0.385124i 0.5 + 2.23871i 0.5 − 1.75780i 0.5 + 0.385124i
−1.13090 1.95878i 0 −1.55787 + 2.69832i −1.00000 0 0.630901 1.09275i 2.52360 0 1.13090 + 1.95878i
406.2 −0.169938 0.294342i 0 0.942242 1.63201i −1.00000 0 −0.330062 + 0.571683i −1.32025 0 0.169938 + 0.294342i
406.3 1.30084 + 2.25312i 0 −2.38437 + 4.12985i −1.00000 0 −1.80084 + 3.11915i −7.20336 0 −1.30084 2.25312i
451.1 −1.13090 + 1.95878i 0 −1.55787 2.69832i −1.00000 0 0.630901 + 1.09275i 2.52360 0 1.13090 1.95878i
451.2 −0.169938 + 0.294342i 0 0.942242 + 1.63201i −1.00000 0 −0.330062 0.571683i −1.32025 0 0.169938 0.294342i
451.3 1.30084 2.25312i 0 −2.38437 4.12985i −1.00000 0 −1.80084 3.11915i −7.20336 0 −1.30084 + 2.25312i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 451.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.j.f 6
3.b odd 2 1 195.2.i.d 6
13.c even 3 1 inner 585.2.j.f 6
13.c even 3 1 7605.2.a.bv 3
13.e even 6 1 7605.2.a.bw 3
15.d odd 2 1 975.2.i.l 6
15.e even 4 2 975.2.bb.k 12
39.h odd 6 1 2535.2.a.ba 3
39.i odd 6 1 195.2.i.d 6
39.i odd 6 1 2535.2.a.bb 3
195.x odd 6 1 975.2.i.l 6
195.bl even 12 2 975.2.bb.k 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.d 6 3.b odd 2 1
195.2.i.d 6 39.i odd 6 1
585.2.j.f 6 1.a even 1 1 trivial
585.2.j.f 6 13.c even 3 1 inner
975.2.i.l 6 15.d odd 2 1
975.2.i.l 6 195.x odd 6 1
975.2.bb.k 12 15.e even 4 2
975.2.bb.k 12 195.bl even 12 2
2535.2.a.ba 3 39.h odd 6 1
2535.2.a.bb 3 39.i odd 6 1
7605.2.a.bv 3 13.c even 3 1
7605.2.a.bw 3 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 6T_{2}^{4} + 4T_{2}^{3} + 36T_{2}^{2} + 12T_{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 6 T^{4} + 4 T^{3} + 36 T^{2} + \cdots + 4$$
$3$ $$T^{6}$$
$5$ $$(T + 1)^{6}$$
$7$ $$T^{6} + 3 T^{5} + 12 T^{4} - 3 T^{3} + \cdots + 9$$
$11$ $$T^{6} + 24 T^{4} - 32 T^{3} + \cdots + 256$$
$13$ $$T^{6} - 3 T^{5} - 6 T^{4} + 83 T^{3} + \cdots + 2197$$
$17$ $$T^{6} + 42 T^{4} + 196 T^{3} + \cdots + 9604$$
$19$ $$T^{6} + 12 T^{5} + 108 T^{4} + \cdots + 16$$
$23$ $$T^{6} + 48 T^{4} - 192 T^{3} + \cdots + 9216$$
$29$ $$T^{6} - 6 T^{5} + 36 T^{4} - 28 T^{3} + \cdots + 196$$
$31$ $$(T^{3} - 3 T^{2} - 93 T + 363)^{2}$$
$37$ $$T^{6} + 6 T^{5} + 72 T^{4} - 232 T^{3} + \cdots + 64$$
$41$ $$T^{6} + 24 T^{4} - 52 T^{3} + \cdots + 676$$
$43$ $$T^{6} + 9 T^{5} + 66 T^{4} + 157 T^{3} + \cdots + 121$$
$47$ $$(T^{3} - 12 T^{2} - 6 T + 206)^{2}$$
$53$ $$(T^{3} - 24 T - 16)^{2}$$
$59$ $$T^{6} + 6 T^{5} + 48 T^{4} - 44 T^{3} + \cdots + 196$$
$61$ $$T^{6} - 3 T^{5} + 30 T^{4} + \cdots + 4489$$
$67$ $$T^{6} + 9 T^{5} + 162 T^{4} + \cdots + 123201$$
$71$ $$T^{6} + 12 T^{5} + 192 T^{4} + \cdots + 465124$$
$73$ $$(T^{3} - 21 T^{2} + 93 T + 89)^{2}$$
$79$ $$(T^{3} + 3 T^{2} - 93 T - 363)^{2}$$
$83$ $$(T^{3} - 18 T^{2} + 60 T + 168)^{2}$$
$89$ $$T^{6} - 30 T^{5} + 612 T^{4} + \cdots + 777924$$
$97$ $$T^{6} + 15 T^{5} + 234 T^{4} + \cdots + 346921$$