# Properties

 Label 546.2.k.d Level $546$ Weight $2$ Character orbit 546.k Analytic conductor $4.360$ Analytic rank $0$ Dimension $8$ 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.k (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.6498455769.2 Defining polynomial: $$x^{8} - x^{7} + 6 x^{6} + 3 x^{5} + 25 x^{4} - 3 x^{3} + 6 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 6 x^{6} + 3 x^{5} + 25 x^{4} - 3 x^{3} + 6 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$6 \nu^{7} + 21 \nu^{6} - 14 \nu^{5} + 193 \nu^{4} + 126 \nu^{3} + 532 \nu^{2} - 664 \nu + 112$$$$)/119$$ $$\beta_{2}$$ $$=$$ $$($$$$8 \nu^{7} - 6 \nu^{6} + 21 \nu^{5} + 59 \nu^{4} + 66 \nu^{3} - 84 \nu^{2} - 449 \nu - 15$$$$)/119$$ $$\beta_{3}$$ $$=$$ $$($$$$-15 \nu^{7} + 7 \nu^{6} - 84 \nu^{5} - 66 \nu^{4} - 434 \nu^{3} - 21 \nu^{2} - 6 \nu + 315$$$$)/119$$ $$\beta_{4}$$ $$=$$ $$($$$$-22 \nu^{7} + 42 \nu^{6} - 147 \nu^{5} + 46 \nu^{4} - 462 \nu^{3} + 588 \nu^{2} - 104 \nu + 105$$$$)/119$$ $$\beta_{5}$$ $$=$$ $$($$$$-24 \nu^{7} + 69 \nu^{6} - 182 \nu^{5} + 180 \nu^{4} - 402 \nu^{3} + 1085 \nu^{2} - 81 \nu - 6$$$$)/119$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{7} - 3 \nu^{6} + 14 \nu^{5} - \nu^{4} + 54 \nu^{3} - 28 \nu^{2} + 47 \nu - 4$$$$)/7$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} - 4 \nu^{6} + 28 \nu^{5} + 22 \nu^{4} + 121 \nu^{3} + 7 \nu^{2} + 2 \nu + 4$$$$)/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + 2 \beta_{4} + 2 \beta_{2} - \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} - 3 \beta_{5} + 7 \beta_{4} + \beta_{2} + \beta_{1} - 6$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{7} + 5 \beta_{6} - 5 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} + 10 \beta_{1} - 9$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{6} + 6 \beta_{5} - 17 \beta_{4} + 6 \beta_{3} - 7 \beta_{2} + 3 \beta_{1} - 6$$ $$\nu^{5}$$ $$=$$ $$($$$$18 \beta_{7} - 64 \beta_{6} + 30 \beta_{5} - 89 \beta_{4} - 50 \beta_{2} - 32 \beta_{1} + 57$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$30 \beta_{7} - 71 \beta_{6} + 71 \beta_{4} - 114 \beta_{3} + 71 \beta_{2} - 142 \beta_{1} + 324$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$217 \beta_{6} - 243 \beta_{5} + 890 \beta_{4} - 243 \beta_{3} + 548 \beta_{2} - 217 \beta_{1} + 243$$$$)/3$$

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-\beta_{4}$$ $$1$$ $$-1 + \beta_{4}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
373.1
 0.271028 − 0.469434i −0.922415 + 1.59767i 1.33821 − 2.31784i −0.186817 + 0.323577i 0.271028 + 0.469434i −0.922415 − 1.59767i 1.33821 + 2.31784i −0.186817 − 0.323577i
0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −1.15139 + 1.99426i 0.500000 + 0.866025i −0.964471 + 2.46370i −1.00000 1.00000 −2.30278
373.2 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −1.15139 + 1.99426i 0.500000 + 0.866025i 2.61586 + 0.396592i −1.00000 1.00000 −2.30278
373.3 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0.651388 1.12824i 0.500000 + 0.866025i −2.36323 + 1.18960i −1.00000 1.00000 1.30278
373.4 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0.651388 1.12824i 0.500000 + 0.866025i 2.21184 1.45181i −1.00000 1.00000 1.30278
445.1 0.500000 0.866025i 1.00000 −0.500000 0.866025i −1.15139 1.99426i 0.500000 0.866025i −0.964471 2.46370i −1.00000 1.00000 −2.30278
445.2 0.500000 0.866025i 1.00000 −0.500000 0.866025i −1.15139 1.99426i 0.500000 0.866025i 2.61586 0.396592i −1.00000 1.00000 −2.30278
445.3 0.500000 0.866025i 1.00000 −0.500000 0.866025i 0.651388 + 1.12824i 0.500000 0.866025i −2.36323 1.18960i −1.00000 1.00000 1.30278
445.4 0.500000 0.866025i 1.00000 −0.500000 0.866025i 0.651388 + 1.12824i 0.500000 0.866025i 2.21184 + 1.45181i −1.00000 1.00000 1.30278
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 445.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.k.d yes 8
3.b odd 2 1 1638.2.p.g 8
7.c even 3 1 546.2.j.b 8
13.c even 3 1 546.2.j.b 8
21.h odd 6 1 1638.2.m.i 8
39.i odd 6 1 1638.2.m.i 8
91.g even 3 1 inner 546.2.k.d yes 8
273.bm odd 6 1 1638.2.p.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.b 8 7.c even 3 1
546.2.j.b 8 13.c even 3 1
546.2.k.d yes 8 1.a even 1 1 trivial
546.2.k.d yes 8 91.g even 3 1 inner
1638.2.m.i 8 21.h odd 6 1
1638.2.m.i 8 39.i odd 6 1
1638.2.p.g 8 3.b odd 2 1
1638.2.p.g 8 273.bm odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + T_{5}^{3} + 4 T_{5}^{2} - 3 T_{5} + 9$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{4}$$
$3$ $$( -1 + T )^{8}$$
$5$ $$( 9 - 3 T + 4 T^{2} + T^{3} + T^{4} )^{2}$$
$7$ $$2401 - 1029 T - 196 T^{2} + 21 T^{3} + 57 T^{4} + 3 T^{5} - 4 T^{6} - 3 T^{7} + T^{8}$$
$11$ $$( 27 + 63 T - 36 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$13$ $$28561 - 15379 T + 3718 T^{2} - 845 T^{3} + 221 T^{4} - 65 T^{5} + 22 T^{6} - 7 T^{7} + T^{8}$$
$17$ $$81 - 189 T + 423 T^{2} - 150 T^{3} + 121 T^{4} + 54 T^{5} + 34 T^{6} + 6 T^{7} + T^{8}$$
$19$ $$( 299 - 104 T - 51 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$23$ $$729 - 1215 T + 1701 T^{2} - 756 T^{3} + 351 T^{4} - 42 T^{5} + 28 T^{6} - 4 T^{7} + T^{8}$$
$29$ $$81 + 189 T + 423 T^{2} + 150 T^{3} + 121 T^{4} - 54 T^{5} + 34 T^{6} - 6 T^{7} + T^{8}$$
$31$ $$9 + 15 T + 103 T^{2} - 190 T^{3} + 629 T^{4} - 250 T^{5} + 74 T^{6} - 10 T^{7} + T^{8}$$
$37$ $$687241 - 554601 T + 493985 T^{2} + 17568 T^{3} + 10335 T^{4} + 666 T^{5} + 200 T^{6} + 12 T^{7} + T^{8}$$
$41$ $$4397409 - 874449 T + 446499 T^{2} + 29046 T^{3} + 17305 T^{4} + 54 T^{5} + 166 T^{6} + 6 T^{7} + T^{8}$$
$43$ $$29800681 - 2232731 T + 1084393 T^{2} + 25040 T^{3} + 24401 T^{4} + 146 T^{5} + 184 T^{6} + 4 T^{7} + T^{8}$$
$47$ $$14085009 + 4458564 T + 1343790 T^{2} + 148986 T^{3} + 24273 T^{4} + 2070 T^{5} + 307 T^{6} + 17 T^{7} + T^{8}$$
$53$ $$1108809 + 739206 T + 644436 T^{2} - 94770 T^{3} + 21789 T^{4} - 972 T^{5} + 153 T^{6} - 3 T^{7} + T^{8}$$
$59$ $$18429849 + 579555 T^{2} + 13932 T^{4} + 135 T^{6} + T^{8}$$
$61$ $$( 848 + 88 T - 144 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$67$ $$( -313 - 329 T - 105 T^{2} - 7 T^{3} + T^{4} )^{2}$$
$71$ $$227529 - 217512 T + 164529 T^{2} - 47220 T^{3} + 11494 T^{4} - 366 T^{5} + 127 T^{6} - 6 T^{7} + T^{8}$$
$73$ $$301401 - 377163 T + 471420 T^{2} - 21549 T^{3} + 12505 T^{4} + 1393 T^{5} + 360 T^{6} + 19 T^{7} + T^{8}$$
$79$ $$860307561 - 144543168 T + 22437331 T^{2} - 1718352 T^{3} + 151572 T^{4} - 8344 T^{5} + 639 T^{6} - 24 T^{7} + T^{8}$$
$83$ $$( 9 + 21 T + 2 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$89$ $$59049 + 91854 T + 128304 T^{2} + 26082 T^{3} + 6489 T^{4} + 336 T^{5} + 109 T^{6} + 7 T^{7} + T^{8}$$
$97$ $$299209 - 66734 T + 100216 T^{2} + 46382 T^{3} + 21833 T^{4} + 3656 T^{5} + 469 T^{6} + 25 T^{7} + T^{8}$$