# Properties

 Label 2760.2.k.c Level $2760$ Weight $2$ Character orbit 2760.k Analytic conductor $22.039$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(2209,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

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

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

chi := DirichletCharacter("2760.2209");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2760.k (of order $$2$$, degree $$1$$, 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: $$22.0387109579$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 31x^{10} + 359x^{8} + 1957x^{6} + 5132x^{4} + 5744x^{2} + 1600$$ x^12 + 31*x^10 + 359*x^8 + 1957*x^6 + 5132*x^4 + 5744*x^2 + 1600 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{3} - \beta_{10} q^{5} + \beta_1 q^{7} - q^{9}+O(q^{10})$$ q + b6 * q^3 - b10 * q^5 + b1 * q^7 - q^9 $$q + \beta_{6} q^{3} - \beta_{10} q^{5} + \beta_1 q^{7} - q^{9} + \beta_{2} q^{11} - \beta_{7} q^{13} + \beta_{4} q^{15} + ( - \beta_{10} - \beta_{9} + \beta_{7} - 2 \beta_{6} + \beta_1) q^{17} + (\beta_{10} - \beta_{9} + \beta_{8} + \beta_{4} + \beta_{3} + 2) q^{19} - \beta_{5} q^{21} - \beta_{6} q^{23} + ( - 2 \beta_{9} - 2 \beta_{6} + \beta_{4} + \beta_{3} + 1) q^{25} - \beta_{6} q^{27} + (\beta_{8} - \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{29} + (\beta_{10} - \beta_{9} - \beta_{8} + 3 \beta_{5} - \beta_{2} - 2) q^{31} + \beta_{11} q^{33} + ( - \beta_{11} - \beta_{8} - \beta_{5}) q^{35} + (\beta_{11} - \beta_{10} - \beta_{9} - \beta_{7} + 2 \beta_{6} - \beta_1) q^{37} - \beta_{8} q^{39} + ( - 2 \beta_{8} + 3 \beta_{5} - \beta_{2} - 4) q^{41} + ( - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{4} + \beta_{3}) q^{43} + \beta_{10} q^{45} + (\beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{6} - 2 \beta_1) q^{47} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 3) q^{49} + (\beta_{8} - \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{51} + ( - \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - \beta_1) q^{53} + (\beta_{8} + \beta_{7} - 2 \beta_{5} - \beta_{2}) q^{55} + (\beta_{10} + \beta_{9} - \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{3}) q^{57} + ( - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{59} + (\beta_{8} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 4) q^{61} - \beta_1 q^{63} + (\beta_{11} - \beta_{7} - 2 \beta_{5} + \beta_{2}) q^{65} + ( - \beta_{11} + 3 \beta_{7} + 4 \beta_{6} - \beta_{4} + \beta_{3} - \beta_1) q^{67} + q^{69} + ( - \beta_{10} + \beta_{9} + \beta_{5} - \beta_{2} + 4) q^{71} + ( - 2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 3 \beta_{7} - \beta_{4} + \beta_{3}) q^{73} + (\beta_{10} + \beta_{9} + \beta_{6} + 2 \beta_{3} + 2) q^{75} + (\beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3}) q^{77} + ( - \beta_{10} + \beta_{9} + 2 \beta_{8} - 4) q^{79} + q^{81} + ( - \beta_{7} - 6 \beta_{6} + \beta_1) q^{83} + ( - 2 \beta_{11} - 2 \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 4) q^{85}+ \cdots - \beta_{2} q^{99}+O(q^{100})$$ q + b6 * q^3 - b10 * q^5 + b1 * q^7 - q^9 + b2 * q^11 - b7 * q^13 + b4 * q^15 + (-b10 - b9 + b7 - 2*b6 + b1) * q^17 + (b10 - b9 + b8 + b4 + b3 + 2) * q^19 - b5 * q^21 - b6 * q^23 + (-2*b9 - 2*b6 + b4 + b3 + 1) * q^25 - b6 * q^27 + (b8 - b5 + b4 + b3 + 2) * q^29 + (b10 - b9 - b8 + 3*b5 - b2 - 2) * q^31 + b11 * q^33 + (-b11 - b8 - b5) * q^35 + (b11 - b10 - b9 - b7 + 2*b6 - b1) * q^37 - b8 * q^39 + (-2*b8 + 3*b5 - b2 - 4) * q^41 + (-b11 + b10 + b9 - b7 - b4 + b3) * q^43 + b10 * q^45 + (b11 - b10 - b9 + 2*b6 - 2*b1) * q^47 + (-b5 + b4 + b3 + 3) * q^49 + (b8 - b5 + b4 + b3 + 2) * q^51 + (-b11 - b10 - b9 - 2*b6 - 2*b4 + 2*b3 - b1) * q^53 + (b8 + b7 - 2*b5 - b2) * q^55 + (b10 + b9 - b7 + 2*b6 - b4 + b3) * q^57 + (-b10 + b9 + b8 + b5 + b4 + b3 + 2*b2 + 2) * q^59 + (b8 + b4 + b3 + 2*b2 - 4) * q^61 - b1 * q^63 + (b11 - b7 - 2*b5 + b2) * q^65 + (-b11 + 3*b7 + 4*b6 - b4 + b3 - b1) * q^67 + q^69 + (-b10 + b9 + b5 - b2 + 4) * q^71 + (-2*b11 - 2*b10 - 2*b9 + 3*b7 - b4 + b3) * q^73 + (b10 + b9 + b6 + 2*b3 + 2) * q^75 + (b11 + b10 + b9 + 2*b6 - 2*b4 + 2*b3) * q^77 + (-b10 + b9 + 2*b8 - 4) * q^79 + q^81 + (-b7 - 6*b6 + b1) * q^83 + (-2*b11 - 2*b9 - b8 + b7 - 2*b6 + b5 - b4 + b3 - b2 - 4) * q^85 + (b10 + b9 - b7 + 2*b6 - b1) * q^87 + (b10 - b9 - 2*b8 - 2*b5 - b4 - b3 - 2*b2 + 2) * q^89 + (b10 - b9 - b8 + 2*b4 + 2*b3) * q^91 + (-b11 + b7 - 2*b6 - b4 + b3 + 3*b1) * q^93 + (-b11 - b10 + 3*b9 + b8 - 2*b6 - b4 + b3 + b2 + 2*b1 - 4) * q^95 + (b11 - b10 - b9 - 3*b7 + 4*b6 - 2*b1) * q^97 - b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{9}+O(q^{10})$$ 12 * q - 12 * q^9 $$12 q - 12 q^{9} - 4 q^{15} + 12 q^{19} - 6 q^{21} + 4 q^{25} + 6 q^{29} - 2 q^{31} - 2 q^{35} + 4 q^{39} - 22 q^{41} + 22 q^{49} + 6 q^{51} - 16 q^{55} + 18 q^{59} - 60 q^{61} - 12 q^{65} + 12 q^{69} + 54 q^{71} + 16 q^{75} - 56 q^{79} + 12 q^{81} - 38 q^{85} + 28 q^{89} - 12 q^{91} - 52 q^{95}+O(q^{100})$$ 12 * q - 12 * q^9 - 4 * q^15 + 12 * q^19 - 6 * q^21 + 4 * q^25 + 6 * q^29 - 2 * q^31 - 2 * q^35 + 4 * q^39 - 22 * q^41 + 22 * q^49 + 6 * q^51 - 16 * q^55 + 18 * q^59 - 60 * q^61 - 12 * q^65 + 12 * q^69 + 54 * q^71 + 16 * q^75 - 56 * q^79 + 12 * q^81 - 38 * q^85 + 28 * q^89 - 12 * q^91 - 52 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 31x^{10} + 359x^{8} + 1957x^{6} + 5132x^{4} + 5744x^{2} + 1600$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{10} - 29\nu^{8} - 291\nu^{6} - 1175\nu^{4} - 1632\nu^{2} - 400 ) / 80$$ (-v^10 - 29*v^8 - 291*v^6 - 1175*v^4 - 1632*v^2 - 400) / 80 $$\beta_{3}$$ $$=$$ $$( - \nu^{11} - 6 \nu^{10} - 27 \nu^{9} - 174 \nu^{8} - 243 \nu^{7} - 1786 \nu^{6} - 793 \nu^{5} - 7730 \nu^{4} - 432 \nu^{3} - 12352 \nu^{2} + 784 \nu - 3360 ) / 320$$ (-v^11 - 6*v^10 - 27*v^9 - 174*v^8 - 243*v^7 - 1786*v^6 - 793*v^5 - 7730*v^4 - 432*v^3 - 12352*v^2 + 784*v - 3360) / 320 $$\beta_{4}$$ $$=$$ $$( \nu^{11} - 6 \nu^{10} + 27 \nu^{9} - 174 \nu^{8} + 243 \nu^{7} - 1786 \nu^{6} + 793 \nu^{5} - 7730 \nu^{4} + 432 \nu^{3} - 12352 \nu^{2} - 784 \nu - 3360 ) / 320$$ (v^11 - 6*v^10 + 27*v^9 - 174*v^8 + 243*v^7 - 1786*v^6 + 793*v^5 - 7730*v^4 + 432*v^3 - 12352*v^2 - 784*v - 3360) / 320 $$\beta_{5}$$ $$=$$ $$( -3\nu^{10} - 87\nu^{8} - 893\nu^{6} - 3865\nu^{4} - 6256\nu^{2} - 2000 ) / 80$$ (-3*v^10 - 87*v^8 - 893*v^6 - 3865*v^4 - 6256*v^2 - 2000) / 80 $$\beta_{6}$$ $$=$$ $$( -5\nu^{11} - 143\nu^{9} - 1447\nu^{7} - 6213\nu^{5} - 10200\nu^{3} - 3696\nu ) / 320$$ (-5*v^11 - 143*v^9 - 1447*v^7 - 6213*v^5 - 10200*v^3 - 3696*v) / 320 $$\beta_{7}$$ $$=$$ $$( -3\nu^{11} - 87\nu^{9} - 893\nu^{7} - 3865\nu^{5} - 6176\nu^{3} - 1520\nu ) / 160$$ (-3*v^11 - 87*v^9 - 893*v^7 - 3865*v^5 - 6176*v^3 - 1520*v) / 160 $$\beta_{8}$$ $$=$$ $$( 6\nu^{10} + 169\nu^{8} + 1676\nu^{6} + 7025\nu^{4} + 11192\nu^{2} + 3600 ) / 80$$ (6*v^10 + 169*v^8 + 1676*v^6 + 7025*v^4 + 11192*v^2 + 3600) / 80 $$\beta_{9}$$ $$=$$ $$( - 2 \nu^{11} - 9 \nu^{10} - 56 \nu^{9} - 251 \nu^{8} - 554 \nu^{7} - 2459 \nu^{6} - 2348 \nu^{5} - 10145 \nu^{4} - 3944 \nu^{3} - 15688 \nu^{2} - 1616 \nu - 4560 ) / 160$$ (-2*v^11 - 9*v^10 - 56*v^9 - 251*v^8 - 554*v^7 - 2459*v^6 - 2348*v^5 - 10145*v^4 - 3944*v^3 - 15688*v^2 - 1616*v - 4560) / 160 $$\beta_{10}$$ $$=$$ $$( - 2 \nu^{11} + 9 \nu^{10} - 56 \nu^{9} + 251 \nu^{8} - 554 \nu^{7} + 2459 \nu^{6} - 2348 \nu^{5} + 10145 \nu^{4} - 3944 \nu^{3} + 15688 \nu^{2} - 1616 \nu + 4560 ) / 160$$ (-2*v^11 + 9*v^10 - 56*v^9 + 251*v^8 - 554*v^7 + 2459*v^6 - 2348*v^5 + 10145*v^4 - 3944*v^3 + 15688*v^2 - 1616*v + 4560) / 160 $$\beta_{11}$$ $$=$$ $$( 9\nu^{11} + 251\nu^{9} + 2459\nu^{7} + 10145\nu^{5} + 15688\nu^{3} + 4560\nu ) / 160$$ (9*v^11 + 251*v^9 + 2459*v^7 + 10145*v^5 + 15688*v^3 + 4560*v) / 160
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - 4$$ -b5 + b4 + b3 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{10} + \beta_{9} + 2\beta_{7} - 4\beta_{6} - 7\beta_1$$ b10 + b9 + 2*b7 - 4*b6 - 7*b1 $$\nu^{4}$$ $$=$$ $$-2\beta_{10} + 2\beta_{9} + 4\beta_{8} + 13\beta_{5} - 11\beta_{4} - 11\beta_{3} + 28$$ -2*b10 + 2*b9 + 4*b8 + 13*b5 - 11*b4 - 11*b3 + 28 $$\nu^{5}$$ $$=$$ $$-4\beta_{11} - 21\beta_{10} - 21\beta_{9} - 26\beta_{7} + 52\beta_{6} + 4\beta_{4} - 4\beta_{3} + 63\beta_1$$ -4*b11 - 21*b10 - 21*b9 - 26*b7 + 52*b6 + 4*b4 - 4*b3 + 63*b1 $$\nu^{6}$$ $$=$$ $$34\beta_{10} - 34\beta_{9} - 68\beta_{8} - 157\beta_{5} + 119\beta_{4} + 119\beta_{3} + 12\beta_{2} - 244$$ 34*b10 - 34*b9 - 68*b8 - 157*b5 + 119*b4 + 119*b3 + 12*b2 - 244 $$\nu^{7}$$ $$=$$ $$80\beta_{11} + 305\beta_{10} + 305\beta_{9} + 306\beta_{7} - 604\beta_{6} - 92\beta_{4} + 92\beta_{3} - 639\beta_1$$ 80*b11 + 305*b10 + 305*b9 + 306*b7 - 604*b6 - 92*b4 + 92*b3 - 639*b1 $$\nu^{8}$$ $$=$$ $$-466\beta_{10} + 466\beta_{9} + 916\beta_{8} + 1853\beta_{5} - 1331\beta_{4} - 1331\beta_{3} - 264\beta_{2} + 2396$$ -466*b10 + 466*b9 + 916*b8 + 1853*b5 - 1331*b4 - 1331*b3 - 264*b2 + 2396 $$\nu^{9}$$ $$=$$ $$- 1196 \beta_{11} - 3949 \beta_{10} - 3949 \beta_{9} - 3578 \beta_{7} + 6884 \beta_{6} + 1444 \beta_{4} - 1444 \beta_{3} + 6911 \beta_1$$ -1196*b11 - 3949*b10 - 3949*b9 - 3578*b7 + 6884*b6 + 1444*b4 - 1444*b3 + 6911*b1 $$\nu^{10}$$ $$=$$ $$5970 \beta_{10} - 5970 \beta_{9} - 11476 \beta_{8} - 21693 \beta_{5} + 15263 \beta_{4} + 15263 \beta_{3} + 4084 \beta_{2} - 25252$$ 5970*b10 - 5970*b9 - 11476*b8 - 21693*b5 + 15263*b4 + 15263*b3 + 4084*b2 - 25252 $$\nu^{11}$$ $$=$$ $$16024 \beta_{11} + 48729 \beta_{10} + 48729 \beta_{9} + 42002 \beta_{7} - 78604 \beta_{6} - 19644 \beta_{4} + 19644 \beta_{3} - 77471 \beta_1$$ 16024*b11 + 48729*b10 + 48729*b9 + 42002*b7 - 78604*b6 - 19644*b4 + 19644*b3 - 77471*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times$$.

 $$n$$ $$1201$$ $$1381$$ $$1657$$ $$1841$$ $$2071$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2209.1
 − 2.70731i 1.70731i − 3.43090i 2.43090i − 1.63473i 0.634730i − 1.70731i 2.70731i − 2.43090i 3.43090i − 0.634730i 1.63473i
0 1.00000i 0 −2.21432 0.311108i 0 2.70731i 0 −1.00000 0
2209.2 0 1.00000i 0 −2.21432 0.311108i 0 1.70731i 0 −1.00000 0
2209.3 0 1.00000i 0 0.539189 2.17009i 0 3.43090i 0 −1.00000 0
2209.4 0 1.00000i 0 0.539189 2.17009i 0 2.43090i 0 −1.00000 0
2209.5 0 1.00000i 0 1.67513 + 1.48119i 0 1.63473i 0 −1.00000 0
2209.6 0 1.00000i 0 1.67513 + 1.48119i 0 0.634730i 0 −1.00000 0
2209.7 0 1.00000i 0 −2.21432 + 0.311108i 0 1.70731i 0 −1.00000 0
2209.8 0 1.00000i 0 −2.21432 + 0.311108i 0 2.70731i 0 −1.00000 0
2209.9 0 1.00000i 0 0.539189 + 2.17009i 0 2.43090i 0 −1.00000 0
2209.10 0 1.00000i 0 0.539189 + 2.17009i 0 3.43090i 0 −1.00000 0
2209.11 0 1.00000i 0 1.67513 1.48119i 0 0.634730i 0 −1.00000 0
2209.12 0 1.00000i 0 1.67513 1.48119i 0 1.63473i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2209.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2760.2.k.c 12
5.b even 2 1 inner 2760.2.k.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.k.c 12 1.a even 1 1 trivial
2760.2.k.c 12 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{12} + 31T_{7}^{10} + 359T_{7}^{8} + 1957T_{7}^{6} + 5132T_{7}^{4} + 5744T_{7}^{2} + 1600$$ acting on $$S_{2}^{\mathrm{new}}(2760, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$(T^{2} + 1)^{6}$$
$5$ $$(T^{6} - T^{4} + 16 T^{3} - 5 T^{2} + 125)^{2}$$
$7$ $$T^{12} + 31 T^{10} + 359 T^{8} + \cdots + 1600$$
$11$ $$(T^{6} - 32 T^{4} - 46 T^{3} + 120 T^{2} + \cdots - 160)^{2}$$
$13$ $$T^{12} + 48 T^{10} + 844 T^{8} + \cdots + 25600$$
$17$ $$T^{12} + 91 T^{10} + 2567 T^{8} + \cdots + 118336$$
$19$ $$(T^{6} - 6 T^{5} - 42 T^{4} + 266 T^{3} + \cdots + 800)^{2}$$
$23$ $$(T^{2} + 1)^{6}$$
$29$ $$(T^{6} - 3 T^{5} - 41 T^{4} + 3 T^{3} + \cdots + 344)^{2}$$
$31$ $$(T^{6} + T^{5} - 141 T^{4} + 195 T^{3} + \cdots - 1600)^{2}$$
$37$ $$T^{12} + 191 T^{10} + \cdots + 30913600$$
$41$ $$(T^{6} + 11 T^{5} - 115 T^{4} - 1645 T^{3} + \cdots - 1336)^{2}$$
$43$ $$T^{12} + 188 T^{10} + 10384 T^{8} + \cdots + 1024$$
$47$ $$T^{12} + 228 T^{10} + 12472 T^{8} + \cdots + 4194304$$
$53$ $$T^{12} + 431 T^{10} + \cdots + 13601344$$
$59$ $$(T^{6} - 9 T^{5} - 145 T^{4} + 1221 T^{3} + \cdots + 5000)^{2}$$
$61$ $$(T^{6} + 30 T^{5} + 246 T^{4} + \cdots + 117200)^{2}$$
$67$ $$T^{12} + 623 T^{10} + \cdots + 217581199936$$
$71$ $$(T^{6} - 27 T^{5} + 217 T^{4} - 251 T^{3} + \cdots - 5608)^{2}$$
$73$ $$T^{12} + 608 T^{10} + \cdots + 89718784$$
$79$ $$(T^{6} + 28 T^{5} + 208 T^{4} + \cdots + 71168)^{2}$$
$83$ $$T^{12} + 295 T^{10} + \cdots + 547185664$$
$89$ $$(T^{6} - 14 T^{5} - 204 T^{4} + \cdots - 405760)^{2}$$
$97$ $$T^{12} + 592 T^{10} + \cdots + 876160000$$