# Properties

 Label 1856.4.a.bg Level $1856$ Weight $4$ Character orbit 1856.a Self dual yes Analytic conductor $109.508$ Analytic rank $1$ Dimension $9$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1856.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$109.507544971$$ Analytic rank: $$1$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802$$ x^9 - 4*x^8 - 138*x^7 + 394*x^6 + 5872*x^5 - 10822*x^4 - 85158*x^3 + 30654*x^2 + 439999*x + 396802 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 928) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{4} - \beta_1 - 1) q^{5} + ( - \beta_{6} - 1) q^{7} + (\beta_{8} + \beta_{5} - \beta_{4} + 2 \beta_1 + 5) q^{9}+O(q^{10})$$ q + b1 * q^3 + (b4 - b1 - 1) * q^5 + (-b6 - 1) * q^7 + (b8 + b5 - b4 + 2*b1 + 5) * q^9 $$q + \beta_1 q^{3} + (\beta_{4} - \beta_1 - 1) q^{5} + ( - \beta_{6} - 1) q^{7} + (\beta_{8} + \beta_{5} - \beta_{4} + 2 \beta_1 + 5) q^{9} + (\beta_{5} - \beta_{4} + \beta_{2} + 7) q^{11} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 6) q^{13} + ( - 2 \beta_{8} + \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{2} - 6 \beta_1 - 18) q^{15} + ( - 2 \beta_{8} + \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + \cdots - 9) q^{17}+ \cdots + (12 \beta_{8} + 2 \beta_{7} + 12 \beta_{6} + 3 \beta_{5} + 29 \beta_{4} + \cdots + 469) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b4 - b1 - 1) * q^5 + (-b6 - 1) * q^7 + (b8 + b5 - b4 + 2*b1 + 5) * q^9 + (b5 - b4 + b2 + 7) * q^11 + (-b6 - b5 - b4 - b3 - b2 - b1 - 6) * q^13 + (-2*b8 + b6 - b5 + 3*b4 - b2 - 6*b1 - 18) * q^15 + (-2*b8 + b7 + 3*b6 - 2*b5 + 2*b4 - b3 + b2 - b1 - 9) * q^17 + (b8 + b7 + b6 - 3*b4 + b3 - 2*b2 + b1 + 5) * q^19 + (-b8 - 2*b6 + b5 - 2*b4 + 3*b3 + b2 - 6*b1 - 6) * q^21 + (2*b8 + 3*b6 - 2*b5 + 4*b4 - b3 + 3*b2 - 3*b1 - 4) * q^23 + (-b8 - 4*b7 - b6 + 2*b5 - 3*b4 + 4*b3 + 2*b2 + 13*b1 + 7) * q^25 + (4*b8 - 2*b7 - b5 - 7*b4 - 2*b3 - b2 + 10*b1 + 37) * q^27 + 29 * q^29 + (3*b8 - b7 + 4*b6 + b5 + 6*b4 - 6*b2 + 4*b1 + 4) * q^31 + (-2*b8 + 3*b7 + 2*b6 - 5*b5 + b4 - 3*b3 - 7*b2 + 19*b1 - 14) * q^33 + (-b8 - 5*b7 + 2*b6 - b5 - 4*b4 + 3*b1 + 56) * q^35 + (2*b8 - 2*b7 - 8*b4 + b3 + 3*b2 - 3*b1 + 11) * q^37 + (b8 - b7 + 3*b5 - 2*b4 + 4*b3 + 8*b2 - 20*b1 - 58) * q^39 + (b7 - b6 - 6*b4 + 8*b3 + 14*b1 - 48) * q^41 + (-5*b8 + 5*b7 + 7*b6 - 8*b5 + 3*b4 + b3 + 2*b2 - 26*b1 + 3) * q^43 + (-13*b8 + 3*b7 + 6*b6 - 4*b5 + 2*b4 + 6*b2 - 54*b1 - 101) * q^45 + (-b8 + 5*b7 + 4*b6 - b4 - b3 - 12*b2 - 20*b1 - 122) * q^47 + (-7*b8 + b7 - 3*b6 - 5*b5 - 6*b4 - 9*b3 + b2 + 16*b1 - 43) * q^49 + (-7*b8 + 11*b7 + 12*b6 - 3*b5 + 16*b4 - 9*b3 - 9*b2 - 48*b1 + 51) * q^51 + (b8 - 7*b7 - 4*b6 + 2*b5 - 21*b4 - 8*b3 - 2*b2 - b1 - 18) * q^53 + (4*b8 + b6 + 6*b5 - 2*b4 - 3*b3 + 5*b2 - 18*b1 - 60) * q^55 + (6*b8 - 6*b7 - 9*b6 + 3*b5 - 14*b4 + 18*b2 + 37*b1 - 4) * q^57 + (-7*b8 + 5*b7 + 2*b6 + b5 + 16*b4 + 16*b3 + 4*b2 - 45*b1 + 76) * q^59 + (-3*b8 + 12*b7 + 20*b6 - 11*b5 + 12*b4 + 4*b3 - 39*b1 - 113) * q^61 + (-10*b8 + 8*b6 - 4*b5 - 2*b4 + 9*b3 + 3*b2 - 29*b1 - 179) * q^63 + (3*b8 + 4*b7 - 2*b6 - 7*b5 - 19*b4 - 8*b3 - 4*b2 + 20*b1 - 76) * q^65 + (-12*b8 - 14*b7 - 2*b6 + 6*b5 - 2*b4 - 5*b3 + 17*b2 - 53*b1 + 153) * q^67 + (9*b8 + 6*b6 - b5 - 18*b4 - 10*b3 - 30*b2 + 27*b1 - 19) * q^69 + (-4*b8 - 12*b6 - 4*b5 - 28*b4 - 20*b3 - 12*b2 - 14*b1 - 196) * q^71 + (18*b7 - 4*b6 + 14*b5 + 4*b4 - 3*b3 - 21*b2 - 9*b1 - 87) * q^73 + (25*b8 - 11*b7 - 21*b6 + 19*b5 - 14*b4 + 9*b3 - 7*b2 + 32*b1 + 352) * q^75 + (b8 + 2*b7 - 14*b6 + b5 - 12*b4 + 10*b3 - 2*b2 + 7*b1 - 137) * q^77 + (-10*b8 - 10*b7 - 3*b6 + b5 - 11*b4 + 10*b3 + 21*b2 - 48*b1 - 304) * q^79 + (15*b8 - 17*b7 - 6*b6 - 8*b5 - 24*b4 + b3 + 5*b2 + 85*b1 + 19) * q^81 + (11*b8 - 7*b7 - 10*b6 + 25*b5 + 6*b4 + 17*b3 + 3*b2 - 50*b1 + 333) * q^83 + (27*b8 + 8*b7 + 3*b5 - 20*b4 - 11*b3 - 9*b2 + 68*b1 - 18) * q^85 + 29*b1 * q^87 + (2*b8 - 9*b7 + 7*b6 + 20*b5 - 2*b4 - 6*b3 - 18*b2 + 4*b1 - 44) * q^89 + (b8 - b7 - 6*b6 - 13*b5 + 6*b4 - 35*b3 - b2 - 2*b1 + 349) * q^91 + (15*b8 - 23*b7 + 11*b6 + 7*b5 + 15*b4 - 7*b3 + 27*b2 + 57*b1 + 165) * q^93 + (-5*b8 + 15*b7 + 15*b6 - 18*b5 + 9*b4 - 10*b3 - 3*b2 - 56*b1 - 472) * q^95 + (-10*b8 + 15*b7 + 15*b6 + 4*b5 + 4*b4 + 13*b3 + 11*b2 - 91*b1 - 229) * q^97 + (12*b8 + 2*b7 + 12*b6 + 3*b5 + 29*b4 + 2*b3 + 23*b2 - 55*b1 + 469) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10})$$ 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9 $$9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + 64 q^{11} - 70 q^{13} - 170 q^{15} - 66 q^{17} + 42 q^{19} - 76 q^{21} - 40 q^{23} + 111 q^{25} + 322 q^{27} + 261 q^{29} + 64 q^{31} - 52 q^{33} + 496 q^{35} + 54 q^{37} - 590 q^{39} - 378 q^{41} - 32 q^{43} - 1046 q^{45} - 1164 q^{47} - 351 q^{49} + 376 q^{51} - 278 q^{53} - 614 q^{55} + 28 q^{57} + 640 q^{59} - 1054 q^{61} - 1660 q^{63} - 708 q^{65} + 1184 q^{67} - 188 q^{69} - 1988 q^{71} - 750 q^{73} + 3126 q^{75} - 1260 q^{77} - 2916 q^{79} + 293 q^{81} + 2832 q^{83} - 56 q^{85} + 116 q^{87} - 370 q^{89} + 3016 q^{91} + 1696 q^{93} - 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100})$$ 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9 + 64 * q^11 - 70 * q^13 - 170 * q^15 - 66 * q^17 + 42 * q^19 - 76 * q^21 - 40 * q^23 + 111 * q^25 + 322 * q^27 + 261 * q^29 + 64 * q^31 - 52 * q^33 + 496 * q^35 + 54 * q^37 - 590 * q^39 - 378 * q^41 - 32 * q^43 - 1046 * q^45 - 1164 * q^47 - 351 * q^49 + 376 * q^51 - 278 * q^53 - 614 * q^55 + 28 * q^57 + 640 * q^59 - 1054 * q^61 - 1660 * q^63 - 708 * q^65 + 1184 * q^67 - 188 * q^69 - 1988 * q^71 - 750 * q^73 + 3126 * q^75 - 1260 * q^77 - 2916 * q^79 + 293 * q^81 + 2832 * q^83 - 56 * q^85 + 116 * q^87 - 370 * q^89 + 3016 * q^91 + 1696 * q^93 - 4412 * q^95 - 2234 * q^97 + 4118 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 2129 \nu^{8} - 11757 \nu^{7} - 264339 \nu^{6} + 1175057 \nu^{5} + 9556105 \nu^{4} - 33394123 \nu^{3} - 97616725 \nu^{2} + 180274991 \nu + 369231982 ) / 4740120$$ (2129*v^8 - 11757*v^7 - 264339*v^6 + 1175057*v^5 + 9556105*v^4 - 33394123*v^3 - 97616725*v^2 + 180274991*v + 369231982) / 4740120 $$\beta_{3}$$ $$=$$ $$( - 39 \nu^{8} + 317 \nu^{7} + 4409 \nu^{6} - 34357 \nu^{5} - 127235 \nu^{4} + 1014083 \nu^{3} + 248575 \nu^{2} - 4511931 \nu - 860902 ) / 83160$$ (-39*v^8 + 317*v^7 + 4409*v^6 - 34357*v^5 - 127235*v^4 + 1014083*v^3 + 248575*v^2 - 4511931*v - 860902) / 83160 $$\beta_{4}$$ $$=$$ $$( 347 \nu^{8} - 2165 \nu^{7} - 43679 \nu^{6} + 235173 \nu^{5} + 1598859 \nu^{4} - 7355187 \nu^{3} - 16395629 \nu^{2} + 46817579 \nu + 78704822 ) / 316008$$ (347*v^8 - 2165*v^7 - 43679*v^6 + 235173*v^5 + 1598859*v^4 - 7355187*v^3 - 16395629*v^2 + 46817579*v + 78704822) / 316008 $$\beta_{5}$$ $$=$$ $$( - 3196 \nu^{8} + 17103 \nu^{7} + 412971 \nu^{6} - 1812133 \nu^{5} - 15686240 \nu^{4} + 55607237 \nu^{3} + 176498495 \nu^{2} - 336420739 \nu - 787744538 ) / 2370060$$ (-3196*v^8 + 17103*v^7 + 412971*v^6 - 1812133*v^5 - 15686240*v^4 + 55607237*v^3 + 176498495*v^2 - 336420739*v - 787744538) / 2370060 $$\beta_{6}$$ $$=$$ $$( - 8339 \nu^{8} + 49617 \nu^{7} + 1041729 \nu^{6} - 5309147 \nu^{5} - 37053595 \nu^{4} + 163032163 \nu^{3} + 342770455 \nu^{2} - 954029561 \nu - 1519251202 ) / 4740120$$ (-8339*v^8 + 49617*v^7 + 1041729*v^6 - 5309147*v^5 - 37053595*v^4 + 163032163*v^3 + 342770455*v^2 - 954029561*v - 1519251202) / 4740120 $$\beta_{7}$$ $$=$$ $$( 1333 \nu^{8} - 6999 \nu^{7} - 171603 \nu^{6} + 734299 \nu^{5} + 6418745 \nu^{4} - 22288601 \nu^{3} - 67067285 \nu^{2} + 132639157 \nu + 259520594 ) / 677160$$ (1333*v^8 - 6999*v^7 - 171603*v^6 + 734299*v^5 + 6418745*v^4 - 22288601*v^3 - 67067285*v^2 + 132639157*v + 259520594) / 677160 $$\beta_{8}$$ $$=$$ $$( 11597 \nu^{8} - 66681 \nu^{7} - 1481127 \nu^{6} + 7151861 \nu^{5} + 55355365 \nu^{4} - 221542279 \nu^{3} - 594191305 \nu^{2} + 1365624923 \nu + 2604377566 ) / 4740120$$ (11597*v^8 - 66681*v^7 - 1481127*v^6 + 7151861*v^5 + 55355365*v^4 - 221542279*v^3 - 594191305*v^2 + 1365624923*v + 2604377566) / 4740120
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} + \beta_{5} - \beta_{4} + 2\beta _1 + 32$$ b8 + b5 - b4 + 2*b1 + 32 $$\nu^{3}$$ $$=$$ $$4\beta_{8} - 2\beta_{7} - \beta_{5} - 7\beta_{4} - 2\beta_{3} - \beta_{2} + 64\beta _1 + 37$$ 4*b8 - 2*b7 - b5 - 7*b4 - 2*b3 - b2 + 64*b1 + 37 $$\nu^{4}$$ $$=$$ $$96\beta_{8} - 17\beta_{7} - 6\beta_{6} + 73\beta_{5} - 105\beta_{4} + \beta_{3} + 5\beta_{2} + 247\beta _1 + 1882$$ 96*b8 - 17*b7 - 6*b6 + 73*b5 - 105*b4 + b3 + 5*b2 + 247*b1 + 1882 $$\nu^{5}$$ $$=$$ $$580 \beta_{8} - 258 \beta_{7} - 96 \beta_{6} + 88 \beta_{5} - 880 \beta_{4} - 114 \beta_{3} - 108 \beta_{2} + 4905 \beta _1 + 5092$$ 580*b8 - 258*b7 - 96*b6 + 88*b5 - 880*b4 - 114*b3 - 108*b2 + 4905*b1 + 5092 $$\nu^{6}$$ $$=$$ $$8663 \beta_{8} - 2414 \beta_{7} - 1284 \beta_{6} + 5625 \beta_{5} - 10253 \beta_{4} + 250 \beta_{3} + 578 \beta_{2} + 27656 \beta _1 + 134664$$ 8663*b8 - 2414*b7 - 1284*b6 + 5625*b5 - 10253*b4 + 250*b3 + 578*b2 + 27656*b1 + 134664 $$\nu^{7}$$ $$=$$ $$64308 \beta_{8} - 26950 \beta_{7} - 14772 \beta_{6} + 19853 \beta_{5} - 92853 \beta_{4} - 5062 \beta_{3} - 8711 \beta_{2} + 409818 \beta _1 + 611739$$ 64308*b8 - 26950*b7 - 14772*b6 + 19853*b5 - 92853*b4 - 5062*b3 - 8711*b2 + 409818*b1 + 611739 $$\nu^{8}$$ $$=$$ $$788310 \beta_{8} - 261219 \beta_{7} - 161082 \beta_{6} + 461973 \beta_{5} - 984441 \beta_{4} + 30147 \beta_{3} + 47367 \beta_{2} + 2891949 \beta _1 + 10714546$$ 788310*b8 - 261219*b7 - 161082*b6 + 461973*b5 - 984441*b4 + 30147*b3 + 47367*b2 + 2891949*b1 + 10714546

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −7.63776 −7.02483 −2.38920 −1.95108 −1.51339 3.39022 4.87969 6.50540 9.74094
0 −7.63776 0 7.65015 0 −5.64550 0 31.3353 0
1.2 0 −7.02483 0 −2.19045 0 −5.10018 0 22.3483 0
1.3 0 −2.38920 0 3.17647 0 28.0947 0 −21.2917 0
1.4 0 −1.95108 0 −18.0628 0 −22.0180 0 −23.1933 0
1.5 0 −1.51339 0 6.33195 0 −1.77839 0 −24.7097 0
1.6 0 3.39022 0 19.3767 0 7.92881 0 −15.5064 0
1.7 0 4.87969 0 −10.4780 0 23.3159 0 −3.18861 0
1.8 0 6.50540 0 1.74434 0 −26.0581 0 15.3202 0
1.9 0 9.74094 0 −17.5483 0 −10.7391 0 67.8859 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$29$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.4.a.bg 9
4.b odd 2 1 1856.4.a.bf 9
8.b even 2 1 928.4.a.d 9
8.d odd 2 1 928.4.a.e yes 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.4.a.d 9 8.b even 2 1
928.4.a.e yes 9 8.d odd 2 1
1856.4.a.bf 9 4.b odd 2 1
1856.4.a.bg 9 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1856))$$:

 $$T_{3}^{9} - 4 T_{3}^{8} - 138 T_{3}^{7} + 394 T_{3}^{6} + 5872 T_{3}^{5} - 10822 T_{3}^{4} - 85158 T_{3}^{3} + 30654 T_{3}^{2} + 439999 T_{3} + 396802$$ T3^9 - 4*T3^8 - 138*T3^7 + 394*T3^6 + 5872*T3^5 - 10822*T3^4 - 85158*T3^3 + 30654*T3^2 + 439999*T3 + 396802 $$T_{5}^{9} + 10 T_{5}^{8} - 568 T_{5}^{7} - 4532 T_{5}^{6} + 85426 T_{5}^{5} + 256400 T_{5}^{4} - 4664192 T_{5}^{3} + 7153684 T_{5}^{2} + 21340421 T_{5} - 37835002$$ T5^9 + 10*T5^8 - 568*T5^7 - 4532*T5^6 + 85426*T5^5 + 256400*T5^4 - 4664192*T5^3 + 7153684*T5^2 + 21340421*T5 - 37835002 $$T_{7}^{9} + 12 T_{7}^{8} - 1296 T_{7}^{7} - 17936 T_{7}^{6} + 407984 T_{7}^{5} + 6862272 T_{7}^{4} + 2377728 T_{7}^{3} - 333385728 T_{7}^{2} - 1487601664 T_{7} - 1638662144$$ T7^9 + 12*T7^8 - 1296*T7^7 - 17936*T7^6 + 407984*T7^5 + 6862272*T7^4 + 2377728*T7^3 - 333385728*T7^2 - 1487601664*T7 - 1638662144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{9}$$
$3$ $$T^{9} - 4 T^{8} - 138 T^{7} + \cdots + 396802$$
$5$ $$T^{9} + 10 T^{8} - 568 T^{7} + \cdots - 37835002$$
$7$ $$T^{9} + 12 T^{8} + \cdots - 1638662144$$
$11$ $$T^{9} - 64 T^{8} + \cdots - 86950571182$$
$13$ $$T^{9} + 70 T^{8} + \cdots + 119297288098$$
$17$ $$T^{9} + \cdots + 239296714517504$$
$19$ $$T^{9} - 42 T^{8} + \cdots - 71\!\cdots\!16$$
$23$ $$T^{9} + 40 T^{8} + \cdots + 60\!\cdots\!04$$
$29$ $$(T - 29)^{9}$$
$31$ $$T^{9} - 64 T^{8} + \cdots - 12\!\cdots\!34$$
$37$ $$T^{9} - 54 T^{8} + \cdots - 25\!\cdots\!12$$
$41$ $$T^{9} + 378 T^{8} + \cdots + 71\!\cdots\!04$$
$43$ $$T^{9} + 32 T^{8} + \cdots - 30\!\cdots\!42$$
$47$ $$T^{9} + 1164 T^{8} + \cdots - 20\!\cdots\!22$$
$53$ $$T^{9} + 278 T^{8} + \cdots + 25\!\cdots\!86$$
$59$ $$T^{9} - 640 T^{8} + \cdots + 70\!\cdots\!64$$
$61$ $$T^{9} + 1054 T^{8} + \cdots - 19\!\cdots\!72$$
$67$ $$T^{9} - 1184 T^{8} + \cdots - 33\!\cdots\!96$$
$71$ $$T^{9} + 1988 T^{8} + \cdots - 55\!\cdots\!76$$
$73$ $$T^{9} + 750 T^{8} + \cdots - 61\!\cdots\!76$$
$79$ $$T^{9} + 2916 T^{8} + \cdots + 16\!\cdots\!86$$
$83$ $$T^{9} - 2832 T^{8} + \cdots + 77\!\cdots\!36$$
$89$ $$T^{9} + 370 T^{8} + \cdots - 37\!\cdots\!00$$
$97$ $$T^{9} + 2234 T^{8} + \cdots + 21\!\cdots\!88$$