# Properties

 Label 462.2.j.f Level $462$ Weight $2$ Character orbit 462.j Analytic conductor $3.689$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.j (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.64000000.1 Defining polynomial: $$x^{8} - 2 x^{6} + 4 x^{4} - 8 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{6} + 4 x^{4} - 8 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4}$$$$/4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5}$$$$/4$$ $$\beta_{6}$$ $$=$$ $$\nu^{6}$$$$/8$$ $$\beta_{7}$$ $$=$$ $$\nu^{7}$$$$/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{6}$$ $$\nu^{7}$$ $$=$$ $$8 \beta_{7}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/462\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 −0.831254 + 1.14412i 0.831254 − 1.14412i 1.34500 + 0.437016i −1.34500 − 0.437016i 1.34500 − 0.437016i −1.34500 + 0.437016i −0.831254 − 1.14412i 0.831254 + 1.14412i
−0.809017 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i −2.65401 + 1.92825i 0.809017 0.587785i −0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i 3.28054
169.2 −0.809017 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i 0.0359800 0.0261410i 0.809017 0.587785i −0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i −0.0444738
295.1 0.309017 0.951057i 0.809017 0.587785i −0.809017 0.587785i −1.02224 3.14612i −0.309017 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i −3.30803
295.2 0.309017 0.951057i 0.809017 0.587785i −0.809017 0.587785i 0.640271 + 1.97055i −0.309017 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i 2.07196
379.1 0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i −1.02224 + 3.14612i −0.309017 + 0.951057i 0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i −3.30803
379.2 0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i 0.640271 1.97055i −0.309017 + 0.951057i 0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i 2.07196
421.1 −0.809017 + 0.587785i −0.309017 0.951057i 0.309017 0.951057i −2.65401 1.92825i 0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i 3.28054
421.2 −0.809017 + 0.587785i −0.309017 0.951057i 0.309017 0.951057i 0.0359800 + 0.0261410i 0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i −0.0444738
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 421.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.j.f 8
11.c even 5 1 inner 462.2.j.f 8
11.c even 5 1 5082.2.a.cc 4
11.d odd 10 1 5082.2.a.bx 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.j.f 8 1.a even 1 1 trivial
462.2.j.f 8 11.c even 5 1 inner
5082.2.a.bx 4 11.d odd 10 1
5082.2.a.cc 4 11.c even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$3$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$5$ $$1 - 36 T + 492 T^{2} + 182 T^{3} + 150 T^{4} + 68 T^{5} + 27 T^{6} + 6 T^{7} + T^{8}$$
$7$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$11$ $$14641 - 18634 T + 12705 T^{2} - 5896 T^{3} + 2039 T^{4} - 536 T^{5} + 105 T^{6} - 14 T^{7} + T^{8}$$
$13$ $$26896 - 5248 T + 5048 T^{2} + 464 T^{3} + 140 T^{4} - 56 T^{5} + 38 T^{6} - 8 T^{7} + T^{8}$$
$17$ $$78961 - 20794 T + 3657 T^{2} + 748 T^{3} + 390 T^{4} + 22 T^{5} + 52 T^{6} + 4 T^{7} + T^{8}$$
$19$ $$72361 - 62408 T + 27038 T^{2} - 4886 T^{3} + 1040 T^{4} + 104 T^{5} + 23 T^{6} + 2 T^{7} + T^{8}$$
$23$ $$( 451 + 2 T - 51 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$29$ $$1628176 + 438944 T + 41232 T^{2} - 8048 T^{3} + 5840 T^{4} - 752 T^{5} + 102 T^{6} - 4 T^{7} + T^{8}$$
$31$ $$7425625 + 1635000 T + 695750 T^{2} + 31250 T^{3} + 3400 T^{4} - 200 T^{5} + 55 T^{6} + 10 T^{7} + T^{8}$$
$37$ $$3025 + 550 T + 4075 T^{2} + 3700 T^{3} + 2190 T^{4} + 790 T^{5} + 180 T^{6} + 20 T^{7} + T^{8}$$
$41$ $$1957201 - 444882 T + 267513 T^{2} - 2724 T^{3} + 1750 T^{4} + 126 T^{5} + 68 T^{6} - 12 T^{7} + T^{8}$$
$43$ $$( -9644 + 1616 T + 76 T^{2} - 24 T^{3} + T^{4} )^{2}$$
$47$ $$400 - 1600 T^{2} + 16840 T^{4} + 210 T^{6} + T^{8}$$
$53$ $$839056 - 630208 T + 170688 T^{2} + 23024 T^{3} + 6600 T^{4} + 944 T^{5} + 158 T^{6} + 12 T^{7} + T^{8}$$
$59$ $$913936 - 298272 T + 106128 T^{2} - 35664 T^{3} + 9040 T^{4} - 1104 T^{5} + 118 T^{6} - 12 T^{7} + T^{8}$$
$61$ $$274576 + 192832 T + 243008 T^{2} + 46464 T^{3} + 13640 T^{4} + 3264 T^{5} + 458 T^{6} + 32 T^{7} + T^{8}$$
$67$ $$( -324 - 216 T + 126 T^{2} + 24 T^{3} + T^{4} )^{2}$$
$71$ $$( 1936 + 792 T + 124 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$73$ $$400 + 1600 T^{2} + 2440 T^{4} - 30 T^{6} + T^{8}$$
$79$ $$31584400 - 17084800 T + 5008200 T^{2} - 953600 T^{3} + 130440 T^{4} - 12640 T^{5} + 880 T^{6} - 40 T^{7} + T^{8}$$
$83$ $$28344976 + 12181312 T + 3074368 T^{2} + 497024 T^{3} + 64200 T^{4} + 5824 T^{5} + 538 T^{6} + 32 T^{7} + T^{8}$$
$89$ $$( 241 + 74 T - 29 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$97$ $$633616 + 503072 T + 1185008 T^{2} - 64576 T^{3} + 16400 T^{4} + 784 T^{5} - 22 T^{6} - 8 T^{7} + T^{8}$$