# Properties

 Label 845.2.m.f Level $845$ Weight $2$ Character orbit 845.m Analytic conductor $6.747$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.m (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + \beta_{3} - \beta_1) q^{2} - \beta_{4} q^{3} + ( - 2 \beta_{7} + 2 \beta_{4} - \beta_{2} + 1) q^{4} + \beta_{3} q^{5} + (\beta_{6} - 2 \beta_1) q^{6} + ( - 2 \beta_{6} + 2 \beta_1) q^{7} + ( - \beta_{6} + \beta_{5} + 3 \beta_{3}) q^{8} + ( - \beta_{2} + 1) q^{9}+O(q^{10})$$ q + (b5 + b3 - b1) * q^2 - b4 * q^3 + (-2*b7 + 2*b4 - b2 + 1) * q^4 + b3 * q^5 + (b6 - 2*b1) * q^6 + (-2*b6 + 2*b1) * q^7 + (-b6 + b5 + 3*b3) * q^8 + (-b2 + 1) * q^9 $$q + (\beta_{5} + \beta_{3} - \beta_1) q^{2} - \beta_{4} q^{3} + ( - 2 \beta_{7} + 2 \beta_{4} - \beta_{2} + 1) q^{4} + \beta_{3} q^{5} + (\beta_{6} - 2 \beta_1) q^{6} + ( - 2 \beta_{6} + 2 \beta_1) q^{7} + ( - \beta_{6} + \beta_{5} + 3 \beta_{3}) q^{8} + ( - \beta_{2} + 1) q^{9} + (\beta_{4} - \beta_{2}) q^{10} + (\beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{11} + ( - \beta_{7} + 4) q^{12} + (4 \beta_{7} - 6) q^{14} + \beta_{5} q^{15} - 3 \beta_{2} q^{16} + ( - 2 \beta_{7} + 2 \beta_{4} + 2 \beta_{2} - 2) q^{17} + ( - \beta_{6} + \beta_{5} + \beta_{3}) q^{18} + ( - \beta_{6} - 2 \beta_1) q^{19} + ( - 2 \beta_{6} + \beta_1) q^{20} + ( - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{3}) q^{21} + ( - 3 \beta_{7} + 3 \beta_{4} - 4 \beta_{2} + 4) q^{22} - \beta_{4} q^{23} + (3 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{24} - q^{25} - 4 \beta_{7} q^{27} + ( - 6 \beta_{5} - 10 \beta_{3} + 10 \beta_1) q^{28} - 4 \beta_{4} q^{29} + ( - \beta_{7} + \beta_{4} - 2 \beta_{2} + 2) q^{30} + (3 \beta_{6} - 3 \beta_{5} + 6 \beta_{3}) q^{31} + ( - \beta_{6} - 3 \beta_1) q^{32} + (2 \beta_{6} - 2 \beta_1) q^{33} + 2 \beta_{3} q^{34} + (2 \beta_{7} - 2 \beta_{4} + 2 \beta_{2} - 2) q^{35} + (2 \beta_{4} - \beta_{2}) q^{36} - 6 \beta_{5} q^{37} - \beta_{7} q^{38} + (\beta_{7} - 3) q^{40} + ( - 2 \beta_{5} + 6 \beta_{3} - 6 \beta_1) q^{41} + (6 \beta_{4} - 8 \beta_{2}) q^{42} + (5 \beta_{7} - 5 \beta_{4} + 4 \beta_{2} - 4) q^{43} + ( - 5 \beta_{6} + 5 \beta_{5} + 6 \beta_{3}) q^{44} + \beta_1 q^{45} + (\beta_{6} - 2 \beta_1) q^{46} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3}) q^{47} + ( - 3 \beta_{7} + 3 \beta_{4}) q^{48} + ( - 8 \beta_{4} + 5 \beta_{2}) q^{49} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{50} + (2 \beta_{7} + 4) q^{51} + ( - 6 \beta_{7} - 6) q^{53} + (4 \beta_{5} + 8 \beta_{3} - 8 \beta_1) q^{54} + (\beta_{4} - 2 \beta_{2}) q^{55} + (8 \beta_{7} - 8 \beta_{4} + 10 \beta_{2} - 10) q^{56} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3}) q^{57} + (4 \beta_{6} - 8 \beta_1) q^{58} + (3 \beta_{6} + 6 \beta_1) q^{59} + ( - \beta_{6} + \beta_{5} + 4 \beta_{3}) q^{60} + ( - 8 \beta_{2} + 8) q^{61} + 3 \beta_{4} q^{62} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{63} + ( - 2 \beta_{7} + 7) q^{64} + ( - 4 \beta_{7} + 6) q^{66} + (2 \beta_{3} - 2 \beta_1) q^{67} + ( - 2 \beta_{4} - 6 \beta_{2}) q^{68} + (2 \beta_{2} - 2) q^{69} + (4 \beta_{6} - 4 \beta_{5} - 6 \beta_{3}) q^{70} + (7 \beta_{6} - 2 \beta_1) q^{71} + ( - \beta_{6} + 3 \beta_1) q^{72} + (6 \beta_{6} - 6 \beta_{5}) q^{73} + (6 \beta_{7} - 6 \beta_{4} + 12 \beta_{2} - 12) q^{74} + \beta_{4} q^{75} + (3 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{76} + (6 \beta_{7} - 8) q^{77} + 6 \beta_{7} q^{79} + ( - 3 \beta_{3} + 3 \beta_1) q^{80} + 5 \beta_{2} q^{81} + ( - 4 \beta_{7} + 4 \beta_{4} - 2 \beta_{2} + 2) q^{82} + ( - 2 \beta_{6} + 2 \beta_{5} - 6 \beta_{3}) q^{83} + ( - 10 \beta_{6} + 12 \beta_1) q^{84} + ( - 2 \beta_{6} - 2 \beta_1) q^{85} + (9 \beta_{6} - 9 \beta_{5} - 14 \beta_{3}) q^{86} + (8 \beta_{2} - 8) q^{87} + (5 \beta_{4} - 8 \beta_{2}) q^{88} + (6 \beta_{3} - 6 \beta_1) q^{89} + (\beta_{7} - 1) q^{90} + ( - \beta_{7} + 4) q^{92} + (6 \beta_{5} - 6 \beta_{3} + 6 \beta_1) q^{93} + (4 \beta_{4} - 6 \beta_{2}) q^{94} + (\beta_{7} - \beta_{4} - 2 \beta_{2} + 2) q^{95} + (3 \beta_{6} - 3 \beta_{5} + 2 \beta_{3}) q^{96} + ( - 4 \beta_{6} + 2 \beta_1) q^{97} + (13 \beta_{6} - 21 \beta_1) q^{98} + ( - \beta_{6} + \beta_{5} + 2 \beta_{3}) q^{99}+O(q^{100})$$ q + (b5 + b3 - b1) * q^2 - b4 * q^3 + (-2*b7 + 2*b4 - b2 + 1) * q^4 + b3 * q^5 + (b6 - 2*b1) * q^6 + (-2*b6 + 2*b1) * q^7 + (-b6 + b5 + 3*b3) * q^8 + (-b2 + 1) * q^9 + (b4 - b2) * q^10 + (b5 + 2*b3 - 2*b1) * q^11 + (-b7 + 4) * q^12 + (4*b7 - 6) * q^14 + b5 * q^15 - 3*b2 * q^16 + (-2*b7 + 2*b4 + 2*b2 - 2) * q^17 + (-b6 + b5 + b3) * q^18 + (-b6 - 2*b1) * q^19 + (-2*b6 + b1) * q^20 + (-2*b6 + 2*b5 + 4*b3) * q^21 + (-3*b7 + 3*b4 - 4*b2 + 4) * q^22 - b4 * q^23 + (3*b5 + 2*b3 - 2*b1) * q^24 - q^25 - 4*b7 * q^27 + (-6*b5 - 10*b3 + 10*b1) * q^28 - 4*b4 * q^29 + (-b7 + b4 - 2*b2 + 2) * q^30 + (3*b6 - 3*b5 + 6*b3) * q^31 + (-b6 - 3*b1) * q^32 + (2*b6 - 2*b1) * q^33 + 2*b3 * q^34 + (2*b7 - 2*b4 + 2*b2 - 2) * q^35 + (2*b4 - b2) * q^36 - 6*b5 * q^37 - b7 * q^38 + (b7 - 3) * q^40 + (-2*b5 + 6*b3 - 6*b1) * q^41 + (6*b4 - 8*b2) * q^42 + (5*b7 - 5*b4 + 4*b2 - 4) * q^43 + (-5*b6 + 5*b5 + 6*b3) * q^44 + b1 * q^45 + (b6 - 2*b1) * q^46 + (-2*b6 + 2*b5 + 2*b3) * q^47 + (-3*b7 + 3*b4) * q^48 + (-8*b4 + 5*b2) * q^49 + (-b5 - b3 + b1) * q^50 + (2*b7 + 4) * q^51 + (-6*b7 - 6) * q^53 + (4*b5 + 8*b3 - 8*b1) * q^54 + (b4 - 2*b2) * q^55 + (8*b7 - 8*b4 + 10*b2 - 10) * q^56 + (2*b6 - 2*b5 + 2*b3) * q^57 + (4*b6 - 8*b1) * q^58 + (3*b6 + 6*b1) * q^59 + (-b6 + b5 + 4*b3) * q^60 + (-8*b2 + 8) * q^61 + 3*b4 * q^62 + (-2*b5 - 2*b3 + 2*b1) * q^63 + (-2*b7 + 7) * q^64 + (-4*b7 + 6) * q^66 + (2*b3 - 2*b1) * q^67 + (-2*b4 - 6*b2) * q^68 + (2*b2 - 2) * q^69 + (4*b6 - 4*b5 - 6*b3) * q^70 + (7*b6 - 2*b1) * q^71 + (-b6 + 3*b1) * q^72 + (6*b6 - 6*b5) * q^73 + (6*b7 - 6*b4 + 12*b2 - 12) * q^74 + b4 * q^75 + (3*b5 - 2*b3 + 2*b1) * q^76 + (6*b7 - 8) * q^77 + 6*b7 * q^79 + (-3*b3 + 3*b1) * q^80 + 5*b2 * q^81 + (-4*b7 + 4*b4 - 2*b2 + 2) * q^82 + (-2*b6 + 2*b5 - 6*b3) * q^83 + (-10*b6 + 12*b1) * q^84 + (-2*b6 - 2*b1) * q^85 + (9*b6 - 9*b5 - 14*b3) * q^86 + (8*b2 - 8) * q^87 + (5*b4 - 8*b2) * q^88 + (6*b3 - 6*b1) * q^89 + (b7 - 1) * q^90 + (-b7 + 4) * q^92 + (6*b5 - 6*b3 + 6*b1) * q^93 + (4*b4 - 6*b2) * q^94 + (b7 - b4 - 2*b2 + 2) * q^95 + (3*b6 - 3*b5 + 2*b3) * q^96 + (-4*b6 + 2*b1) * q^97 + (13*b6 - 21*b1) * q^98 + (-b6 + b5 + 2*b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4} + 4 q^{9}+O(q^{10})$$ 8 * q + 4 * q^4 + 4 * q^9 $$8 q + 4 q^{4} + 4 q^{9} - 4 q^{10} + 32 q^{12} - 48 q^{14} - 12 q^{16} - 8 q^{17} + 16 q^{22} - 8 q^{25} + 8 q^{30} - 8 q^{35} - 4 q^{36} - 24 q^{40} - 32 q^{42} - 16 q^{43} + 20 q^{49} + 32 q^{51} - 48 q^{53} - 8 q^{55} - 40 q^{56} + 32 q^{61} + 56 q^{64} + 48 q^{66} - 24 q^{68} - 8 q^{69} - 48 q^{74} - 64 q^{77} + 20 q^{81} + 8 q^{82} - 32 q^{87} - 32 q^{88} - 8 q^{90} + 32 q^{92} - 24 q^{94} + 8 q^{95}+O(q^{100})$$ 8 * q + 4 * q^4 + 4 * q^9 - 4 * q^10 + 32 * q^12 - 48 * q^14 - 12 * q^16 - 8 * q^17 + 16 * q^22 - 8 * q^25 + 8 * q^30 - 8 * q^35 - 4 * q^36 - 24 * q^40 - 32 * q^42 - 16 * q^43 + 20 * q^49 + 32 * q^51 - 48 * q^53 - 8 * q^55 - 40 * q^56 + 32 * q^61 + 56 * q^64 + 48 * q^66 - 24 * q^68 - 8 * q^69 - 48 * q^74 - 64 * q^77 + 20 * q^81 + 8 * q^82 - 32 * q^87 - 32 * q^88 - 8 * q^90 + 32 * q^92 - 24 * q^94 + 8 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}$$ v^7 + v $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}$$ -v^7 + v $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3}$$ -v^7 + v^5 + v^3 $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -v^5 + v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{5} + \beta_{4} ) / 2$$ (b5 + b4) / 2 $$\zeta_{24}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} ) / 2$$ (b7 + b6 - b5) / 2 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2$$ (-b7 + b6 + b4) / 2 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{5} + \beta_{4} ) / 2$$ (-b5 + b4) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
316.1
 −0.965926 + 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 − 0.965926i
−2.09077 1.20711i 0.707107 1.22474i 1.91421 + 3.31552i 1.00000i −2.95680 + 1.70711i 4.18154 2.41421i 4.41421i 0.500000 + 0.866025i −1.20711 + 2.09077i
316.2 −0.358719 0.207107i −0.707107 + 1.22474i −0.914214 1.58346i 1.00000i 0.507306 0.292893i 0.717439 0.414214i 1.58579i 0.500000 + 0.866025i 0.207107 0.358719i
316.3 0.358719 + 0.207107i −0.707107 + 1.22474i −0.914214 1.58346i 1.00000i −0.507306 + 0.292893i −0.717439 + 0.414214i 1.58579i 0.500000 + 0.866025i 0.207107 0.358719i
316.4 2.09077 + 1.20711i 0.707107 1.22474i 1.91421 + 3.31552i 1.00000i 2.95680 1.70711i −4.18154 + 2.41421i 4.41421i 0.500000 + 0.866025i −1.20711 + 2.09077i
361.1 −2.09077 + 1.20711i 0.707107 + 1.22474i 1.91421 3.31552i 1.00000i −2.95680 1.70711i 4.18154 + 2.41421i 4.41421i 0.500000 0.866025i −1.20711 2.09077i
361.2 −0.358719 + 0.207107i −0.707107 1.22474i −0.914214 + 1.58346i 1.00000i 0.507306 + 0.292893i 0.717439 + 0.414214i 1.58579i 0.500000 0.866025i 0.207107 + 0.358719i
361.3 0.358719 0.207107i −0.707107 1.22474i −0.914214 + 1.58346i 1.00000i −0.507306 0.292893i −0.717439 0.414214i 1.58579i 0.500000 0.866025i 0.207107 + 0.358719i
361.4 2.09077 1.20711i 0.707107 + 1.22474i 1.91421 3.31552i 1.00000i 2.95680 + 1.70711i −4.18154 2.41421i 4.41421i 0.500000 0.866025i −1.20711 2.09077i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.m.f 8
13.b even 2 1 inner 845.2.m.f 8
13.c even 3 1 845.2.c.b 4
13.c even 3 1 inner 845.2.m.f 8
13.d odd 4 1 845.2.e.c 4
13.d odd 4 1 845.2.e.h 4
13.e even 6 1 845.2.c.b 4
13.e even 6 1 inner 845.2.m.f 8
13.f odd 12 1 65.2.a.b 2
13.f odd 12 1 845.2.a.g 2
13.f odd 12 1 845.2.e.c 4
13.f odd 12 1 845.2.e.h 4
39.k even 12 1 585.2.a.m 2
39.k even 12 1 7605.2.a.x 2
52.l even 12 1 1040.2.a.j 2
65.o even 12 1 325.2.b.f 4
65.s odd 12 1 325.2.a.i 2
65.s odd 12 1 4225.2.a.r 2
65.t even 12 1 325.2.b.f 4
91.bc even 12 1 3185.2.a.j 2
104.u even 12 1 4160.2.a.z 2
104.x odd 12 1 4160.2.a.bf 2
143.o even 12 1 7865.2.a.j 2
156.v odd 12 1 9360.2.a.cd 2
195.bc odd 12 1 2925.2.c.r 4
195.bh even 12 1 2925.2.a.u 2
195.bn odd 12 1 2925.2.c.r 4
260.bc even 12 1 5200.2.a.bu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 13.f odd 12 1
325.2.a.i 2 65.s odd 12 1
325.2.b.f 4 65.o even 12 1
325.2.b.f 4 65.t even 12 1
585.2.a.m 2 39.k even 12 1
845.2.a.g 2 13.f odd 12 1
845.2.c.b 4 13.c even 3 1
845.2.c.b 4 13.e even 6 1
845.2.e.c 4 13.d odd 4 1
845.2.e.c 4 13.f odd 12 1
845.2.e.h 4 13.d odd 4 1
845.2.e.h 4 13.f odd 12 1
845.2.m.f 8 1.a even 1 1 trivial
845.2.m.f 8 13.b even 2 1 inner
845.2.m.f 8 13.c even 3 1 inner
845.2.m.f 8 13.e even 6 1 inner
1040.2.a.j 2 52.l even 12 1
2925.2.a.u 2 195.bh even 12 1
2925.2.c.r 4 195.bc odd 12 1
2925.2.c.r 4 195.bn odd 12 1
3185.2.a.j 2 91.bc even 12 1
4160.2.a.z 2 104.u even 12 1
4160.2.a.bf 2 104.x odd 12 1
4225.2.a.r 2 65.s odd 12 1
5200.2.a.bu 2 260.bc even 12 1
7605.2.a.x 2 39.k even 12 1
7865.2.a.j 2 143.o even 12 1
9360.2.a.cd 2 156.v odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 6T_{2}^{6} + 35T_{2}^{4} - 6T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 6 T^{6} + 35 T^{4} - 6 T^{2} + \cdots + 1$$
$3$ $$(T^{4} + 2 T^{2} + 4)^{2}$$
$5$ $$(T^{2} + 1)^{4}$$
$7$ $$T^{8} - 24 T^{6} + 560 T^{4} + \cdots + 256$$
$11$ $$T^{8} - 12 T^{6} + 140 T^{4} + \cdots + 16$$
$13$ $$T^{8}$$
$17$ $$(T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16)^{2}$$
$19$ $$T^{8} - 12 T^{6} + 140 T^{4} + \cdots + 16$$
$23$ $$(T^{4} + 2 T^{2} + 4)^{2}$$
$29$ $$(T^{4} + 32 T^{2} + 1024)^{2}$$
$31$ $$(T^{4} + 108 T^{2} + 324)^{2}$$
$37$ $$(T^{4} - 72 T^{2} + 5184)^{2}$$
$41$ $$T^{8} - 88 T^{6} + 6960 T^{4} + \cdots + 614656$$
$43$ $$(T^{4} + 8 T^{3} + 98 T^{2} - 272 T + 1156)^{2}$$
$47$ $$(T^{4} + 24 T^{2} + 16)^{2}$$
$53$ $$(T^{2} + 12 T - 36)^{4}$$
$59$ $$T^{8} - 108 T^{6} + 11340 T^{4} + \cdots + 104976$$
$61$ $$(T^{2} - 8 T + 64)^{4}$$
$67$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$71$ $$T^{8} - 204 T^{6} + \cdots + 78074896$$
$73$ $$(T^{2} + 72)^{4}$$
$79$ $$(T^{2} - 72)^{4}$$
$83$ $$(T^{4} + 88 T^{2} + 784)^{2}$$
$89$ $$(T^{4} - 36 T^{2} + 1296)^{2}$$
$97$ $$T^{8} - 72 T^{6} + 4400 T^{4} + \cdots + 614656$$