# Properties

 Label 378.4.f.b Level $378$ Weight $4$ Character orbit 378.f Analytic conductor $22.303$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 378.f (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.3027219822$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2x^{7} - 2x^{6} + 53x^{5} + 38x^{4} - 166x^{3} + 7x^{2} + 1543x + 2707$$ x^8 - 2*x^7 - 2*x^6 + 53*x^5 + 38*x^4 - 166*x^3 + 7*x^2 + 1543*x + 2707 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{6}$$ Twist minimal: no (minimal twist has level 126) 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{2} + 2) q^{2} - 4 \beta_{2} q^{4} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{5} + ( - 7 \beta_{2} + 7) q^{7} - 8 q^{8}+O(q^{10})$$ q + (-2*b2 + 2) * q^2 - 4*b2 * q^4 + (b7 - b6 - b5 - b4 - b3 + b1) * q^5 + (-7*b2 + 7) * q^7 - 8 * q^8 $$q + ( - 2 \beta_{2} + 2) q^{2} - 4 \beta_{2} q^{4} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{5} + ( - 7 \beta_{2} + 7) q^{7} - 8 q^{8} + ( - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{10} + (5 \beta_{7} - 3 \beta_{6} - \beta_{5} + \beta_{2} - 1) q^{11} + ( - 2 \beta_{7} - 3 \beta_{6} - \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + \beta_1) q^{13} - 14 \beta_{2} q^{14} + (16 \beta_{2} - 16) q^{16} + (\beta_{4} + 3 \beta_{3} + 6 \beta_1 + 9) q^{17} + ( - 4 \beta_{4} + 11 \beta_{3} - 3 \beta_1 - 26) q^{19} + ( - 4 \beta_{7} + 4 \beta_{6} + 4 \beta_{5}) q^{20} + (10 \beta_{7} - 6 \beta_{6} - 2 \beta_{5} - 6 \beta_{4} - 10 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{22} + (6 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} - 94 \beta_{2} - 3 \beta_1) q^{23} + ( - 6 \beta_{7} + 7 \beta_{6} + 16 \beta_{5} + 4 \beta_{2} - 4) q^{25} + ( - 6 \beta_{4} + 4 \beta_{3} + 2 \beta_1 + 10) q^{26} - 28 q^{28} + (8 \beta_{7} - 7 \beta_{6} - 6 \beta_{5} + 69 \beta_{2} - 69) q^{29} + ( - 11 \beta_{7} - 9 \beta_{6} + 13 \beta_{5} - 9 \beta_{4} + 11 \beta_{3} + \cdots - 13 \beta_1) q^{31}+ \cdots - 98 q^{98}+O(q^{100})$$ q + (-2*b2 + 2) * q^2 - 4*b2 * q^4 + (b7 - b6 - b5 - b4 - b3 + b1) * q^5 + (-7*b2 + 7) * q^7 - 8 * q^8 + (-2*b4 - 2*b3 + 2*b1) * q^10 + (5*b7 - 3*b6 - b5 + b2 - 1) * q^11 + (-2*b7 - 3*b6 - b5 - 3*b4 + 2*b3 + 5*b2 + b1) * q^13 - 14*b2 * q^14 + (16*b2 - 16) * q^16 + (b4 + 3*b3 + 6*b1 + 9) * q^17 + (-4*b4 + 11*b3 - 3*b1 - 26) * q^19 + (-4*b7 + 4*b6 + 4*b5) * q^20 + (10*b7 - 6*b6 - 2*b5 - 6*b4 - 10*b3 + 2*b2 + 2*b1) * q^22 + (6*b7 + 4*b6 + 3*b5 + 4*b4 - 6*b3 - 94*b2 - 3*b1) * q^23 + (-6*b7 + 7*b6 + 16*b5 + 4*b2 - 4) * q^25 + (-6*b4 + 4*b3 + 2*b1 + 10) * q^26 - 28 * q^28 + (8*b7 - 7*b6 - 6*b5 + 69*b2 - 69) * q^29 + (-11*b7 - 9*b6 + 13*b5 - 9*b4 + 11*b3 + 52*b2 - 13*b1) * q^31 + 32*b2 * q^32 + (6*b7 - 2*b6 + 12*b5 - 18*b2 + 18) * q^34 + (-7*b4 - 7*b3 + 7*b1) * q^35 + (-25*b4 + 15*b3 + 23*b1 - 40) * q^37 + (22*b7 + 8*b6 - 6*b5 + 52*b2 - 52) * q^38 + (-8*b7 + 8*b6 + 8*b5 + 8*b4 + 8*b3 - 8*b1) * q^40 + (-31*b7 + 9*b6 + 25*b5 + 9*b4 + 31*b3 + 43*b2 - 25*b1) * q^41 + (51*b7 - 6*b6 - 10*b5 - 56*b2 + 56) * q^43 + (-12*b4 - 20*b3 + 4*b1 + 4) * q^44 + (8*b4 - 12*b3 - 6*b1 - 188) * q^46 + (5*b7 + 9*b6 + 30*b5 + 156*b2 - 156) * q^47 - 49*b2 * q^49 + (-12*b7 + 14*b6 + 32*b5 + 14*b4 + 12*b3 + 8*b2 - 32*b1) * q^50 + (8*b7 + 12*b6 + 4*b5 - 20*b2 + 20) * q^52 + (-40*b4 - 22*b3 - 5*b1 - 26) * q^53 + (-4*b4 + 21*b3 + 35*b1 - 337) * q^55 + (56*b2 - 56) * q^56 + (16*b7 - 14*b6 - 12*b5 - 14*b4 - 16*b3 + 138*b2 + 12*b1) * q^58 + (56*b7 - 8*b6 - 7*b5 - 8*b4 - 56*b3 - 49*b2 + 7*b1) * q^59 + (44*b7 - 14*b6 + 7*b5 - 310*b2 + 310) * q^61 + (-18*b4 + 22*b3 - 26*b1 + 104) * q^62 + 64 * q^64 + (-17*b7 - 7*b6 + 3*b5 + 134*b2 - 134) * q^65 + (47*b7 - 29*b6 - 38*b5 - 29*b4 - 47*b3 - 23*b2 + 38*b1) * q^67 + (12*b7 - 4*b6 + 24*b5 - 4*b4 - 12*b3 - 36*b2 - 24*b1) * q^68 + (-14*b7 + 14*b6 + 14*b5) * q^70 + (5*b4 - 52*b3 - 8*b1 + 321) * q^71 + (-34*b4 - 55*b3 + 25*b1 - 496) * q^73 + (30*b7 + 50*b6 + 46*b5 + 80*b2 - 80) * q^74 + (44*b7 + 16*b6 - 12*b5 + 16*b4 - 44*b3 + 104*b2 + 12*b1) * q^76 + (35*b7 - 21*b6 - 7*b5 - 21*b4 - 35*b3 + 7*b2 + 7*b1) * q^77 + (-81*b7 - 7*b6 - 8*b5 - 384*b2 + 384) * q^79 + (16*b4 + 16*b3 - 16*b1) * q^80 + (18*b4 + 62*b3 - 50*b1 + 86) * q^82 + (29*b7 + 52*b6 - 43*b5 - b2 + 1) * q^83 + (-88*b7 + 45*b6 + 37*b5 + 45*b4 + 88*b3 + 191*b2 - 37*b1) * q^85 + (102*b7 - 12*b6 - 20*b5 - 12*b4 - 102*b3 - 112*b2 + 20*b1) * q^86 + (-40*b7 + 24*b6 + 8*b5 - 8*b2 + 8) * q^88 + (56*b4 + 147*b3 - 28*b1 + 33) * q^89 + (-21*b4 + 14*b3 + 7*b1 + 35) * q^91 + (-24*b7 - 16*b6 - 12*b5 + 376*b2 - 376) * q^92 + (10*b7 + 18*b6 + 60*b5 + 18*b4 - 10*b3 + 312*b2 - 60*b1) * q^94 + (-49*b7 - 24*b6 - 40*b5 - 24*b4 + 49*b3 - 297*b2 + 40*b1) * q^95 + (38*b7 - 140*b6 - 23*b5 - 183*b2 + 183) * q^97 - 98 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8}+O(q^{10})$$ 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8 $$8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8} - 4 q^{10} - 5 q^{11} + 21 q^{13} - 56 q^{14} - 64 q^{16} + 46 q^{17} - 188 q^{19} - 4 q^{20} + 10 q^{22} - 374 q^{23} - 41 q^{25} + 84 q^{26} - 224 q^{28} - 271 q^{29} + 243 q^{31} + 128 q^{32} + 46 q^{34} - 14 q^{35} - 362 q^{37} - 188 q^{38} + 8 q^{40} + 213 q^{41} + 238 q^{43} + 40 q^{44} - 1496 q^{46} - 675 q^{47} - 196 q^{49} + 82 q^{50} + 84 q^{52} - 108 q^{53} - 2828 q^{55} - 224 q^{56} + 542 q^{58} - 202 q^{59} + 1212 q^{61} + 972 q^{62} + 512 q^{64} - 549 q^{65} - 139 q^{67} - 92 q^{68} - 14 q^{70} + 2590 q^{71} - 4000 q^{73} - 362 q^{74} + 376 q^{76} + 35 q^{77} + 1545 q^{79} + 32 q^{80} + 852 q^{82} + 142 q^{83} + 793 q^{85} - 476 q^{86} + 40 q^{88} + 264 q^{89} + 294 q^{91} - 1496 q^{92} + 1350 q^{94} - 1244 q^{95} + 638 q^{97} - 784 q^{98}+O(q^{100})$$ 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8 - 4 * q^10 - 5 * q^11 + 21 * q^13 - 56 * q^14 - 64 * q^16 + 46 * q^17 - 188 * q^19 - 4 * q^20 + 10 * q^22 - 374 * q^23 - 41 * q^25 + 84 * q^26 - 224 * q^28 - 271 * q^29 + 243 * q^31 + 128 * q^32 + 46 * q^34 - 14 * q^35 - 362 * q^37 - 188 * q^38 + 8 * q^40 + 213 * q^41 + 238 * q^43 + 40 * q^44 - 1496 * q^46 - 675 * q^47 - 196 * q^49 + 82 * q^50 + 84 * q^52 - 108 * q^53 - 2828 * q^55 - 224 * q^56 + 542 * q^58 - 202 * q^59 + 1212 * q^61 + 972 * q^62 + 512 * q^64 - 549 * q^65 - 139 * q^67 - 92 * q^68 - 14 * q^70 + 2590 * q^71 - 4000 * q^73 - 362 * q^74 + 376 * q^76 + 35 * q^77 + 1545 * q^79 + 32 * q^80 + 852 * q^82 + 142 * q^83 + 793 * q^85 - 476 * q^86 + 40 * q^88 + 264 * q^89 + 294 * q^91 - 1496 * q^92 + 1350 * q^94 - 1244 * q^95 + 638 * q^97 - 784 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} - 2x^{6} + 53x^{5} + 38x^{4} - 166x^{3} + 7x^{2} + 1543x + 2707$$ :

 $$\beta_{1}$$ $$=$$ $$( -45\nu^{7} - 152\nu^{6} + 809\nu^{5} - 1570\nu^{4} - 15799\nu^{3} + 13322\nu^{2} + 104222\nu - 75818 ) / 22491$$ (-45*v^7 - 152*v^6 + 809*v^5 - 1570*v^4 - 15799*v^3 + 13322*v^2 + 104222*v - 75818) / 22491 $$\beta_{2}$$ $$=$$ $$( 956 \nu^{7} - 4527 \nu^{6} + 9062 \nu^{5} + 27893 \nu^{4} - 46669 \nu^{3} - 58960 \nu^{2} + 109080 \nu + 1139585 ) / 247401$$ (956*v^7 - 4527*v^6 + 9062*v^5 + 27893*v^4 - 46669*v^3 - 58960*v^2 + 109080*v + 1139585) / 247401 $$\beta_{3}$$ $$=$$ $$( -65\nu^{7} + 314\nu^{6} - 394\nu^{5} - 3814\nu^{4} + 7091\nu^{3} + 2915\nu^{2} - 19694\nu - 97401 ) / 14553$$ (-65*v^7 + 314*v^6 - 394*v^5 - 3814*v^4 + 7091*v^3 + 2915*v^2 - 19694*v - 97401) / 14553 $$\beta_{4}$$ $$=$$ $$( - 2066 \nu^{7} + 4665 \nu^{6} - 491 \nu^{5} - 89666 \nu^{4} + 16261 \nu^{3} + 115456 \nu^{2} + 337857 \nu - 635990 ) / 247401$$ (-2066*v^7 + 4665*v^6 - 491*v^5 - 89666*v^4 + 16261*v^3 + 115456*v^2 + 337857*v - 635990) / 247401 $$\beta_{5}$$ $$=$$ $$( 872\nu^{7} - 4184\nu^{6} + 10588\nu^{5} + 23098\nu^{4} - 47180\nu^{3} + 23353\nu^{2} + 234695\nu + 890847 ) / 35343$$ (872*v^7 - 4184*v^6 + 10588*v^5 + 23098*v^4 - 47180*v^3 + 23353*v^2 + 234695*v + 890847) / 35343 $$\beta_{6}$$ $$=$$ $$( - 8335 \nu^{7} + 36450 \nu^{6} - 74695 \nu^{5} - 269326 \nu^{4} + 276374 \nu^{3} + 446765 \nu^{2} - 2126268 \nu - 10262284 ) / 247401$$ (-8335*v^7 + 36450*v^6 - 74695*v^5 - 269326*v^4 + 276374*v^3 + 446765*v^2 - 2126268*v - 10262284) / 247401 $$\beta_{7}$$ $$=$$ $$( 9432 \nu^{7} - 40945 \nu^{6} + 80167 \nu^{5} + 302416 \nu^{4} - 345233 \nu^{3} - 780263 \nu^{2} + 1668493 \nu + 10687202 ) / 247401$$ (9432*v^7 - 40945*v^6 + 80167*v^5 + 302416*v^4 - 345233*v^3 - 780263*v^2 + 1668493*v + 10687202) / 247401
 $$\nu$$ $$=$$ $$( -\beta_{7} - 2\beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} + 2\beta _1 + 3 ) / 9$$ (-b7 - 2*b6 - b5 - b4 + 2*b3 + 2*b1 + 3) / 9 $$\nu^{2}$$ $$=$$ $$( -3\beta_{7} - 2\beta_{6} - \beta_{4} + 10\beta_{2} - 2 ) / 3$$ (-3*b7 - 2*b6 - b4 + 10*b2 - 2) / 3 $$\nu^{3}$$ $$=$$ $$( -13\beta_{7} - 17\beta_{6} - 4\beta_{5} + 11\beta_{4} - 4\beta_{3} + 18\beta_{2} - 13\beta _1 - 168 ) / 9$$ (-13*b7 - 17*b6 - 4*b5 + 11*b4 - 4*b3 + 18*b2 - 13*b1 - 168) / 9 $$\nu^{4}$$ $$=$$ $$( 12\beta_{7} + 8\beta_{6} + 9\beta_{5} + 19\beta_{4} - 39\beta_{3} - 121\beta_{2} - 21\beta _1 - 139 ) / 3$$ (12*b7 + 8*b6 + 9*b5 + 19*b4 - 39*b3 - 121*b2 - 21*b1 - 139) / 3 $$\nu^{5}$$ $$=$$ $$( 401\beta_{7} + 334\beta_{6} + 230\beta_{5} + 233\beta_{4} - 241\beta_{3} - 2394\beta_{2} - 205\beta _1 + 57 ) / 9$$ (401*b7 + 334*b6 + 230*b5 + 233*b4 - 241*b3 - 2394*b2 - 205*b1 + 57) / 9 $$\nu^{6}$$ $$=$$ $$157\beta_{7} + 131\beta_{6} + 73\beta_{5} - 20\beta_{4} + 7\beta_{3} - 906\beta_{2} + 4\beta _1 + 994$$ 157*b7 + 131*b6 + 73*b5 - 20*b4 + 7*b3 - 906*b2 + 4*b1 + 994 $$\nu^{7}$$ $$=$$ $$( 764\beta_{7} + 745\beta_{6} + 62\beta_{5} - 4258\beta_{4} + 5573\beta_{3} - 270\beta_{2} + 3089\beta _1 + 34347 ) / 9$$ (764*b7 + 745*b6 + 62*b5 - 4258*b4 + 5573*b3 - 270*b2 + 3089*b1 + 34347) / 9

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 −1.30108 − 1.53317i 2.17384 − 1.62361i 2.52060 + 3.08349i −2.39336 + 0.0732839i −1.30108 + 1.53317i 2.17384 + 1.62361i 2.52060 − 3.08349i −2.39336 − 0.0732839i
1.00000 1.73205i 0 −2.00000 3.46410i −9.75052 16.8884i 0 3.50000 6.06218i −8.00000 0 −39.0021
127.2 1.00000 1.73205i 0 −2.00000 3.46410i 0.721413 + 1.24952i 0 3.50000 6.06218i −8.00000 0 2.88565
127.3 1.00000 1.73205i 0 −2.00000 3.46410i 2.98310 + 5.16688i 0 3.50000 6.06218i −8.00000 0 11.9324
127.4 1.00000 1.73205i 0 −2.00000 3.46410i 5.54600 + 9.60595i 0 3.50000 6.06218i −8.00000 0 22.1840
253.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −9.75052 + 16.8884i 0 3.50000 + 6.06218i −8.00000 0 −39.0021
253.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 0.721413 1.24952i 0 3.50000 + 6.06218i −8.00000 0 2.88565
253.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.98310 5.16688i 0 3.50000 + 6.06218i −8.00000 0 11.9324
253.4 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 5.54600 9.60595i 0 3.50000 + 6.06218i −8.00000 0 22.1840
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.4.f.b 8
3.b odd 2 1 126.4.f.b 8
9.c even 3 1 inner 378.4.f.b 8
9.c even 3 1 1134.4.a.l 4
9.d odd 6 1 126.4.f.b 8
9.d odd 6 1 1134.4.a.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.f.b 8 3.b odd 2 1
126.4.f.b 8 9.d odd 6 1
378.4.f.b 8 1.a even 1 1 trivial
378.4.f.b 8 9.c even 3 1 inner
1134.4.a.l 4 9.c even 3 1
1134.4.a.o 4 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + T_{5}^{7} + 271T_{5}^{6} - 3620T_{5}^{5} + 73087T_{5}^{4} - 448526T_{5}^{3} + 2302885T_{5}^{2} - 3118850T_{5} + 3467044$$ acting on $$S_{4}^{\mathrm{new}}(378, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 4)^{4}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + T^{7} + 271 T^{6} + \cdots + 3467044$$
$7$ $$(T^{2} - 7 T + 49)^{4}$$
$11$ $$T^{8} + 5 T^{7} + \cdots + 97721886025$$
$13$ $$T^{8} - 21 T^{7} + \cdots + 6395840676$$
$17$ $$(T^{4} - 23 T^{3} - 9759 T^{2} + \cdots + 20106721)^{2}$$
$19$ $$(T^{4} + 94 T^{3} - 13617 T^{2} + \cdots + 36369301)^{2}$$
$23$ $$T^{8} + 374 T^{7} + \cdots + 16\!\cdots\!00$$
$29$ $$T^{8} + 271 T^{7} + \cdots + 39\!\cdots\!44$$
$31$ $$T^{8} - 243 T^{7} + \cdots + 14\!\cdots\!00$$
$37$ $$(T^{4} + 181 T^{3} - 159768 T^{2} + \cdots - 2370848354)^{2}$$
$41$ $$T^{8} - 213 T^{7} + \cdots + 28\!\cdots\!09$$
$43$ $$T^{8} - 238 T^{7} + \cdots + 35\!\cdots\!49$$
$47$ $$T^{8} + 675 T^{7} + \cdots + 50\!\cdots\!96$$
$53$ $$(T^{4} + 54 T^{3} - 358917 T^{2} + \cdots + 7638055812)^{2}$$
$59$ $$T^{8} + 202 T^{7} + \cdots + 14\!\cdots\!61$$
$61$ $$T^{8} - 1212 T^{7} + \cdots + 10\!\cdots\!56$$
$67$ $$T^{8} + 139 T^{7} + \cdots + 58\!\cdots\!01$$
$71$ $$(T^{4} - 1295 T^{3} + \cdots - 17389723904)^{2}$$
$73$ $$(T^{4} + 2000 T^{3} + \cdots - 41833011071)^{2}$$
$79$ $$T^{8} - 1545 T^{7} + \cdots + 36\!\cdots\!36$$
$83$ $$T^{8} - 142 T^{7} + \cdots + 28\!\cdots\!76$$
$89$ $$(T^{4} - 132 T^{3} + \cdots - 127304987850)^{2}$$
$97$ $$T^{8} - 638 T^{7} + \cdots + 42\!\cdots\!69$$