# Properties

 Label 546.2.q.g Level $546$ Weight $2$ Character orbit 546.q Analytic conductor $4.360$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (\beta_{3} - \beta_1) q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{3} + \beta_{2} - \beta_1) q^{5} + \beta_{3} q^{6} + (3 \beta_{2} - 1) q^{7} - q^{8} + ( - \beta_{3} + \beta_1 - 3) q^{9}+O(q^{10})$$ q + b2 * q^2 + (b3 - b1) * q^3 + (b2 - 1) * q^4 + (b3 + b2 - b1) * q^5 + b3 * q^6 + (3*b2 - 1) * q^7 - q^8 + (-b3 + b1 - 3) * q^9 $$q + \beta_{2} q^{2} + (\beta_{3} - \beta_1) q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{3} + \beta_{2} - \beta_1) q^{5} + \beta_{3} q^{6} + (3 \beta_{2} - 1) q^{7} - q^{8} + ( - \beta_{3} + \beta_1 - 3) q^{9} + (\beta_{3} + \beta_{2} - 1) q^{10} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{11} + \beta_1 q^{12} + (\beta_{2} + 3) q^{13} + (2 \beta_{2} - 3) q^{14} + (\beta_1 - 3) q^{15} - \beta_{2} q^{16} + (2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{17} + ( - \beta_{3} - 3 \beta_{2}) q^{18} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{19} + (\beta_1 - 1) q^{20} + (2 \beta_{3} + \beta_1) q^{21} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{22} + (\beta_{3} - \beta_{2} + 3) q^{23} + ( - \beta_{3} + \beta_1) q^{24} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{25} + (4 \beta_{2} - 1) q^{26} + ( - 2 \beta_{3} + 2 \beta_1 + 3) q^{27} + ( - \beta_{2} - 2) q^{28} + (\beta_{3} + 1) q^{29} + ( - \beta_{3} - 3 \beta_{2} + \beta_1) q^{30} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{31} + ( - \beta_{2} + 1) q^{32} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{33} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{34} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{35} + ( - 3 \beta_{2} - \beta_1 + 3) q^{36} + ( - 6 \beta_{3} - 2 \beta_{2} - 2) q^{37} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{38} + (4 \beta_{3} - 3 \beta_1) q^{39} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{40} + ( - \beta_{3} + 4 \beta_{2} - 9) q^{41} + ( - \beta_{3} + 3 \beta_1) q^{42} + ( - 4 \beta_{2} + 4) q^{43} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{44} + ( - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{45} + (2 \beta_{2} + \beta_1 + 1) q^{46} + (\beta_{3} + 3 \beta_{2} - \beta_1 - 1) q^{47} - \beta_{3} q^{48} + (3 \beta_{2} - 8) q^{49} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{50} + ( - \beta_{3} - 3 \beta_{2} - \beta_1 - 3) q^{51} + (3 \beta_{2} - 4) q^{52} + (5 \beta_{3} + 7 \beta_{2} - 5 \beta_1 - 1) q^{53} + ( - 2 \beta_{3} + 3 \beta_{2}) q^{54} + ( - 2 \beta_{2} + 4) q^{55} + ( - 3 \beta_{2} + 1) q^{56} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 3) q^{57} + (\beta_{2} + \beta_1) q^{58} + ( - 4 \beta_{2} + \beta_1 - 5) q^{59} + ( - \beta_{3} - 3 \beta_{2} + 3) q^{60} + (3 \beta_{2} + 3) q^{61} + ( - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{62} + ( - 2 \beta_{3} - 9 \beta_{2} - \beta_1 + 3) q^{63} + q^{64} + (4 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 1) q^{65} + (\beta_{3} + 3 \beta_{2} + \beta_1 + 3) q^{66} + (6 \beta_{3} + 2 \beta_{2} + 2) q^{67} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{68} + (\beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{69} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{70} + ( - 6 \beta_{3} + 6 \beta_{2} + 3 \beta_1 - 9) q^{71} + (\beta_{3} - \beta_1 + 3) q^{72} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{73} + ( - 4 \beta_{2} - 6 \beta_1 + 2) q^{74} + (\beta_{3} - 6 \beta_{2} - 2 \beta_1 + 3) q^{75} + (\beta_{3} - 2 \beta_1 + 1) q^{76} + ( - 5 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{77} + (3 \beta_{3} + \beta_1) q^{78} + ( - \beta_{3} - \beta_{2} - \beta_1 + 13) q^{79} + ( - \beta_{3} - \beta_{2} + 1) q^{80} + (5 \beta_{3} - 5 \beta_1 + 6) q^{81} + ( - 5 \beta_{2} - \beta_1 - 4) q^{82} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{83} + ( - 3 \beta_{3} + 2 \beta_1) q^{84} + ( - 2 \beta_{2} - 2) q^{85} + 4 q^{86} + ( - 3 \beta_{2} - \beta_1) q^{87} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{88} + ( - 7 \beta_{3} - 4 \beta_{2} + 1) q^{89} + ( - 2 \beta_{3} - \beta_1 + 3) q^{90} + (11 \beta_{2} - 6) q^{91} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{92} + ( - 12 \beta_{2} - 2 \beta_1 + 6) q^{93} + (\beta_{3} + 2 \beta_{2} - 3) q^{94} + (2 \beta_{2} - 2 \beta_1 + 4) q^{95} - \beta_1 q^{96} + ( - 4 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 4) q^{97} + ( - 5 \beta_{2} - 3) q^{98} + (\beta_{3} - 3 \beta_{2} - 5 \beta_1 - 3) q^{99}+O(q^{100})$$ q + b2 * q^2 + (b3 - b1) * q^3 + (b2 - 1) * q^4 + (b3 + b2 - b1) * q^5 + b3 * q^6 + (3*b2 - 1) * q^7 - q^8 + (-b3 + b1 - 3) * q^9 + (b3 + b2 - 1) * q^10 + (-b3 + 2*b2 + 2*b1 - 1) * q^11 + b1 * q^12 + (b2 + 3) * q^13 + (2*b2 - 3) * q^14 + (b1 - 3) * q^15 - b2 * q^16 + (2*b3 - b2 - b1 + 2) * q^17 + (-b3 - 3*b2) * q^18 + (-2*b3 - b2 + b1) * q^19 + (b1 - 1) * q^20 + (2*b3 + b1) * q^21 + (-2*b3 + b2 + b1 - 2) * q^22 + (b3 - b2 + 3) * q^23 + (-b3 + b1) * q^24 + (b3 + b2 + b1 + 1) * q^25 + (4*b2 - 1) * q^26 + (-2*b3 + 2*b1 + 3) * q^27 + (-b2 - 2) * q^28 + (b3 + 1) * q^29 + (-b3 - 3*b2 + b1) * q^30 + (2*b3 + 2*b2 + 2*b1) * q^31 + (-b2 + 1) * q^32 + (2*b3 - 3*b2 - b1 + 6) * q^33 + (b3 + b2 + b1 + 1) * q^34 + (2*b3 + 2*b2 + b1 - 3) * q^35 + (-3*b2 - b1 + 3) * q^36 + (-6*b3 - 2*b2 - 2) * q^37 + (-b3 - b2 - b1 + 1) * q^38 + (4*b3 - 3*b1) * q^39 + (-b3 - b2 + b1) * q^40 + (-b3 + 4*b2 - 9) * q^41 + (-b3 + 3*b1) * q^42 + (-4*b2 + 4) * q^43 + (-b3 - b2 - b1 - 1) * q^44 + (-3*b3 - 3*b2 + 2*b1 + 3) * q^45 + (2*b2 + b1 + 1) * q^46 + (b3 + 3*b2 - b1 - 1) * q^47 - b3 * q^48 + (3*b2 - 8) * q^49 + (-b3 + 2*b2 + 2*b1 - 1) * q^50 + (-b3 - 3*b2 - b1 - 3) * q^51 + (3*b2 - 4) * q^52 + (5*b3 + 7*b2 - 5*b1 - 1) * q^53 + (-2*b3 + 3*b2) * q^54 + (-2*b2 + 4) * q^55 + (-3*b2 + 1) * q^56 + (b3 + 3*b2 - b1 + 3) * q^57 + (b2 + b1) * q^58 + (-4*b2 + b1 - 5) * q^59 + (-b3 - 3*b2 + 3) * q^60 + (3*b2 + 3) * q^61 + (-2*b3 + 2*b2 + 4*b1 - 2) * q^62 + (-2*b3 - 9*b2 - b1 + 3) * q^63 + q^64 + (4*b3 + 4*b2 - 3*b1 - 1) * q^65 + (b3 + 3*b2 + b1 + 3) * q^66 + (6*b3 + 2*b2 + 2) * q^67 + (-b3 + 2*b2 + 2*b1 - 1) * q^68 + (b3 - 3*b2 - 3*b1) * q^69 + (-b3 - b2 + 3*b1 - 2) * q^70 + (-6*b3 + 6*b2 + 3*b1 - 9) * q^71 + (b3 - b1 + 3) * q^72 + (-2*b3 - 2*b2 - 2*b1 - 4) * q^73 + (-4*b2 - 6*b1 + 2) * q^74 + (b3 - 6*b2 - 2*b1 + 3) * q^75 + (b3 - 2*b1 + 1) * q^76 + (-5*b3 + b2 + b1 - 5) * q^77 + (3*b3 + b1) * q^78 + (-b3 - b2 - b1 + 13) * q^79 + (-b3 - b2 + 1) * q^80 + (5*b3 - 5*b1 + 6) * q^81 + (-5*b2 - b1 - 4) * q^82 + (-2*b3 - 2*b2 + 2*b1) * q^83 + (-3*b3 + 2*b1) * q^84 + (-2*b2 - 2) * q^85 + 4 * q^86 + (-3*b2 - b1) * q^87 + (b3 - 2*b2 - 2*b1 + 1) * q^88 + (-7*b3 - 4*b2 + 1) * q^89 + (-2*b3 - b1 + 3) * q^90 + (11*b2 - 6) * q^91 + (-b3 + 3*b2 + b1 - 2) * q^92 + (-12*b2 - 2*b1 + 6) * q^93 + (b3 + 2*b2 - 3) * q^94 + (2*b2 - 2*b1 + 4) * q^95 - b1 * q^96 + (-4*b3 - 6*b2 + 2*b1 + 4) * q^97 + (-5*b2 - 3) * q^98 + (b3 - 3*b2 - 5*b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - q^{6} + 2 q^{7} - 4 q^{8} - 10 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 - q^6 + 2 * q^7 - 4 * q^8 - 10 * q^9 $$4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - q^{6} + 2 q^{7} - 4 q^{8} - 10 q^{9} - 3 q^{10} + 3 q^{11} + q^{12} + 14 q^{13} - 8 q^{14} - 11 q^{15} - 2 q^{16} + 3 q^{17} - 5 q^{18} + q^{19} - 3 q^{20} - q^{21} - 3 q^{22} + 9 q^{23} + 2 q^{24} + 6 q^{25} + 4 q^{26} + 16 q^{27} - 10 q^{28} + 3 q^{29} - 4 q^{30} + 4 q^{31} + 2 q^{32} + 15 q^{33} + 6 q^{34} - 9 q^{35} + 5 q^{36} - 6 q^{37} + 2 q^{38} - 7 q^{39} - 27 q^{41} + 4 q^{42} + 8 q^{43} - 6 q^{44} + 11 q^{45} + 9 q^{46} + q^{48} - 26 q^{49} + 3 q^{50} - 18 q^{51} - 10 q^{52} + 8 q^{54} + 12 q^{55} - 2 q^{56} + 16 q^{57} + 3 q^{58} - 27 q^{59} + 7 q^{60} + 18 q^{61} + 2 q^{62} - 5 q^{63} + 4 q^{64} - 3 q^{65} + 18 q^{66} + 6 q^{67} + 3 q^{68} - 10 q^{69} - 6 q^{70} - 15 q^{71} + 10 q^{72} - 20 q^{73} - 6 q^{74} - 3 q^{75} + q^{76} - 12 q^{77} - 2 q^{78} + 50 q^{79} + 3 q^{80} + 14 q^{81} - 27 q^{82} + 5 q^{84} - 12 q^{85} + 16 q^{86} - 7 q^{87} - 3 q^{88} + 3 q^{89} + 13 q^{90} - 2 q^{91} - 2 q^{93} - 9 q^{94} + 18 q^{95} - q^{96} + 10 q^{97} - 22 q^{98} - 24 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 - q^6 + 2 * q^7 - 4 * q^8 - 10 * q^9 - 3 * q^10 + 3 * q^11 + q^12 + 14 * q^13 - 8 * q^14 - 11 * q^15 - 2 * q^16 + 3 * q^17 - 5 * q^18 + q^19 - 3 * q^20 - q^21 - 3 * q^22 + 9 * q^23 + 2 * q^24 + 6 * q^25 + 4 * q^26 + 16 * q^27 - 10 * q^28 + 3 * q^29 - 4 * q^30 + 4 * q^31 + 2 * q^32 + 15 * q^33 + 6 * q^34 - 9 * q^35 + 5 * q^36 - 6 * q^37 + 2 * q^38 - 7 * q^39 - 27 * q^41 + 4 * q^42 + 8 * q^43 - 6 * q^44 + 11 * q^45 + 9 * q^46 + q^48 - 26 * q^49 + 3 * q^50 - 18 * q^51 - 10 * q^52 + 8 * q^54 + 12 * q^55 - 2 * q^56 + 16 * q^57 + 3 * q^58 - 27 * q^59 + 7 * q^60 + 18 * q^61 + 2 * q^62 - 5 * q^63 + 4 * q^64 - 3 * q^65 + 18 * q^66 + 6 * q^67 + 3 * q^68 - 10 * q^69 - 6 * q^70 - 15 * q^71 + 10 * q^72 - 20 * q^73 - 6 * q^74 - 3 * q^75 + q^76 - 12 * q^77 - 2 * q^78 + 50 * q^79 + 3 * q^80 + 14 * q^81 - 27 * q^82 + 5 * q^84 - 12 * q^85 + 16 * q^86 - 7 * q^87 - 3 * q^88 + 3 * q^89 + 13 * q^90 - 2 * q^91 - 2 * q^93 - 9 * q^94 + 18 * q^95 - q^96 + 10 * q^97 - 22 * q^98 - 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu + 3 ) / 2$$ (-v^3 + 2*v + 3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3\beta_{2}$$ b3 + 3*b2 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} + 2\beta _1 + 3$$ -2*b3 + 2*b1 + 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/546\mathbb{Z}\right)^\times$$.

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1 - \beta_{2}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −1.18614 + 1.26217i 1.68614 − 0.396143i 1.68614 + 0.396143i −1.18614 − 1.26217i
0.500000 0.866025i −0.500000 1.65831i −0.500000 0.866025i 2.52434i −1.68614 0.396143i 0.500000 2.59808i −1.00000 −2.50000 + 1.65831i −2.18614 1.26217i
251.2 0.500000 0.866025i −0.500000 + 1.65831i −0.500000 0.866025i 0.792287i 1.18614 + 1.26217i 0.500000 2.59808i −1.00000 −2.50000 1.65831i 0.686141 + 0.396143i
335.1 0.500000 + 0.866025i −0.500000 1.65831i −0.500000 + 0.866025i 0.792287i 1.18614 1.26217i 0.500000 + 2.59808i −1.00000 −2.50000 + 1.65831i 0.686141 0.396143i
335.2 0.500000 + 0.866025i −0.500000 + 1.65831i −0.500000 + 0.866025i 2.52434i −1.68614 + 0.396143i 0.500000 + 2.59808i −1.00000 −2.50000 1.65831i −2.18614 + 1.26217i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
273.u even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.q.g yes 4
3.b odd 2 1 546.2.q.e 4
7.b odd 2 1 546.2.q.h yes 4
13.e even 6 1 546.2.q.f yes 4
21.c even 2 1 546.2.q.f yes 4
39.h odd 6 1 546.2.q.h yes 4
91.t odd 6 1 546.2.q.e 4
273.u even 6 1 inner 546.2.q.g yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.q.e 4 3.b odd 2 1
546.2.q.e 4 91.t odd 6 1
546.2.q.f yes 4 13.e even 6 1
546.2.q.f yes 4 21.c even 2 1
546.2.q.g yes 4 1.a even 1 1 trivial
546.2.q.g yes 4 273.u even 6 1 inner
546.2.q.h yes 4 7.b odd 2 1
546.2.q.h yes 4 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5}^{4} + 7T_{5}^{2} + 4$$ T5^4 + 7*T5^2 + 4 $$T_{11}^{4} - 3T_{11}^{3} + 15T_{11}^{2} + 18T_{11} + 36$$ T11^4 - 3*T11^3 + 15*T11^2 + 18*T11 + 36 $$T_{17}^{4} - 3T_{17}^{3} + 15T_{17}^{2} + 18T_{17} + 36$$ T17^4 - 3*T17^3 + 15*T17^2 + 18*T17 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$(T^{2} + T + 3)^{2}$$
$5$ $$T^{4} + 7T^{2} + 4$$
$7$ $$(T^{2} - T + 7)^{2}$$
$11$ $$T^{4} - 3 T^{3} + 15 T^{2} + 18 T + 36$$
$13$ $$(T^{2} - 7 T + 13)^{2}$$
$17$ $$T^{4} - 3 T^{3} + 15 T^{2} + 18 T + 36$$
$19$ $$T^{4} - T^{3} + 9 T^{2} + 8 T + 64$$
$23$ $$T^{4} - 9 T^{3} + 31 T^{2} - 36 T + 16$$
$29$ $$T^{4} - 3 T^{3} + T^{2} + 6 T + 4$$
$31$ $$(T^{2} - 2 T - 32)^{2}$$
$37$ $$T^{4} + 6 T^{3} - 84 T^{2} + \cdots + 9216$$
$41$ $$T^{4} + 27 T^{3} + 301 T^{2} + \cdots + 3364$$
$43$ $$(T^{2} - 4 T + 16)^{2}$$
$47$ $$T^{4} + 19T^{2} + 16$$
$53$ $$T^{4} + 211T^{2} + 1024$$
$59$ $$T^{4} + 27 T^{3} + 301 T^{2} + \cdots + 3364$$
$61$ $$(T^{2} - 9 T + 27)^{2}$$
$67$ $$T^{4} - 6 T^{3} - 84 T^{2} + \cdots + 9216$$
$71$ $$T^{4} + 15 T^{3} + 243 T^{2} + \cdots + 324$$
$73$ $$(T^{2} + 10 T - 8)^{2}$$
$79$ $$(T^{2} - 25 T + 148)^{2}$$
$83$ $$T^{4} + 28T^{2} + 64$$
$89$ $$T^{4} - 3 T^{3} - 131 T^{2} + \cdots + 17956$$
$97$ $$T^{4} - 10 T^{3} + 108 T^{2} + \cdots + 64$$