# Properties

 Label 261.4.a.f Level $261$ Weight $4$ Character orbit 261.a Self dual yes Analytic conductor $15.399$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$261 = 3^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 261.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.3994985115$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.13458092.1 Defining polynomial: $$x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8$$ x^5 - x^4 - 14*x^3 + 18*x^2 + 20*x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 29) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{4} + 2 \beta_{3} + 2 \beta_1 + 6) q^{4} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{5} + ( - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1 + 8) q^{7} + (4 \beta_{4} + 4 \beta_{3} + 8 \beta_{2} + 9 \beta_1 + 16) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b4 + 2*b3 + 2*b1 + 6) * q^4 + (-2*b4 + b3 + b2 - b1 - 3) * q^5 + (-2*b4 - 2*b2 + 2*b1 + 8) * q^7 + (4*b4 + 4*b3 + 8*b2 + 9*b1 + 16) * q^8 $$q + \beta_1 q^{2} + (\beta_{4} + 2 \beta_{3} + 2 \beta_1 + 6) q^{4} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{5} + ( - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1 + 8) q^{7} + (4 \beta_{4} + 4 \beta_{3} + 8 \beta_{2} + 9 \beta_1 + 16) q^{8} + (6 \beta_{4} + 4 \beta_{2} + \beta_1 - 12) q^{10} + ( - 9 \beta_{4} - 5 \beta_{3} - 10 \beta_{2} - 7 \beta_1 - 3) q^{11} + ( - 6 \beta_{4} + 5 \beta_{3} - 9 \beta_{2} - 3 \beta_1 + 5) q^{13} + (4 \beta_{4} + 14 \beta_1 + 40) q^{14} + (9 \beta_{4} + 18 \beta_{3} + 16 \beta_{2} + 30 \beta_1 + 30) q^{16} + ( - 4 \beta_{3} - 10 \beta_{2} - 10) q^{17} + (4 \beta_{4} - 2 \beta_{3} + 8 \beta_1 + 44) q^{19} + (9 \beta_{4} + 2 \beta_{3} - 8 \beta_{2} - 8 \beta_1 + 6) q^{20} + ( - 9 \beta_{4} - 34 \beta_{3} - 20 \beta_{2} - 28 \beta_1 - 22) q^{22} + ( - 24 \beta_{4} - 14 \beta_{3} + 2 \beta_{2} - 46) q^{23} + ( - 2 \beta_{4} - 21 \beta_{3} + 11 \beta_{2} - 17 \beta_1 + 32) q^{25} + (10 \beta_{4} - 24 \beta_{3} + 20 \beta_{2} + 25 \beta_1 - 20) q^{26} + (22 \beta_{4} + 28 \beta_{3} + 16 \beta_{2} + 48 \beta_1 + 116) q^{28} + 29 q^{29} + (5 \beta_{4} + 7 \beta_{3} - 14 \beta_{2} - 19 \beta_1 + 93) q^{31} + (32 \beta_{4} + 60 \beta_{3} + 8 \beta_{2} + 81 \beta_1 + 152) q^{32} + ( - 18 \beta_{4} - 20 \beta_{3} - 16 \beta_{2} - 26 \beta_1 + 36) q^{34} + (30 \beta_{4} + 12 \beta_{3} + 42 \beta_{2} - 14 \beta_1 + 8) q^{35} + ( - 28 \beta_{4} - 22 \beta_{3} + 30 \beta_{2} + 22 \beta_1 + 48) q^{37} + ( - 4 \beta_{4} + 16 \beta_{3} - 8 \beta_{2} + 48 \beta_1 + 104) q^{38} + ( - 78 \beta_{4} - 32 \beta_{3} - 24 \beta_{2} - 19 \beta_1 - 44) q^{40} + ( - 44 \beta_{4} - 22 \beta_{3} - 16 \beta_{2} - 18 \beta_1 + 216) q^{41} + ( - 11 \beta_{4} + 25 \beta_{3} + 22 \beta_{2} - 29 \beta_1 - 49) q^{43} + ( - 26 \beta_{4} - 56 \beta_{3} - 56 \beta_{2} - 149 \beta_1 - 156) q^{44} + (22 \beta_{4} + 4 \beta_{3} - 56 \beta_{2} - 78 \beta_1 + 148) q^{46} + (7 \beta_{4} + 41 \beta_{3} + 16 \beta_{2} - 75 \beta_1 - 45) q^{47} + (4 \beta_{4} - 24 \beta_{3} - 32 \beta_{2} + 40 \beta_1 - 47) q^{49} + ( - 44 \beta_{4} - 12 \beta_{3} - 84 \beta_{2} - 84 \beta_1 - 168) q^{50} + (25 \beta_{4} + 50 \beta_{3} - 24 \beta_{2} - 52 \beta_1 + 326) q^{52} + ( - 10 \beta_{4} - 23 \beta_{3} - 17 \beta_{2} - 75 \beta_1 + 109) q^{53} + (101 \beta_{4} + 41 \beta_{3} + 112 \beta_{2} - 13 \beta_1 + 113) q^{55} + (44 \beta_{4} + 128 \beta_{3} + 112 \beta_{2} + 190 \beta_1 + 120) q^{56} + 29 \beta_1 q^{58} + (16 \beta_{4} - 30 \beta_{3} + 6 \beta_{2} - 52 \beta_1 - 90) q^{59} + (12 \beta_{4} + 60 \beta_{3} + 66 \beta_{2} - 20 \beta_1 + 114) q^{61} + ( - 29 \beta_{4} - 66 \beta_{3} + 28 \beta_{2} + 78 \beta_1 - 286) q^{62} + (73 \beta_{4} + 34 \beta_{3} + 112 \beta_{2} + 282 \beta_1 + 510) q^{64} + (146 \beta_{4} + 15 \beta_{3} + 49 \beta_{2} - 51 \beta_1 + 373) q^{65} + ( - 44 \beta_{3} - 140 \beta_{2} - 88 \beta_1 + 280) q^{67} + ( - 46 \beta_{4} - 52 \beta_{3} - 78 \beta_1 - 100) q^{68} + ( - 8 \beta_{4} + 56 \beta_{3} + 48 \beta_{2} - 2 \beta_1 - 448) q^{70} + ( - 6 \beta_{4} + 122 \beta_{3} + 96 \beta_{2} + 58 \beta_1 + 122) q^{71} + ( - 32 \beta_{4} - 38 \beta_{3} - 66 \beta_{2} - 106 \beta_1 - 384) q^{73} + (64 \beta_{4} + 104 \beta_{3} - 88 \beta_{2} + 32 \beta_1 + 448) q^{74} + (48 \beta_{4} + 96 \beta_{3} + 64 \beta_{2} + 204 \beta_1 + 288) q^{76} + ( - 48 \beta_{4} - 176 \beta_{3} - 158 \beta_{2} - 144 \beta_1 + 332) q^{77} + ( - 177 \beta_{4} - 63 \beta_{3} - 140 \beta_{2} - 139 \beta_1 + 27) q^{79} + ( - 23 \beta_{4} - 102 \beta_{3} - 64 \beta_{2} - 68 \beta_1 + 174) q^{80} + (10 \beta_{4} - 68 \beta_{3} - 88 \beta_{2} + 136 \beta_1 + 44) q^{82} + ( - 60 \beta_{4} - 14 \beta_{3} + 42 \beta_{2} + 96 \beta_1 - 146) q^{83} + (136 \beta_{4} + 66 \beta_{3} + 20 \beta_{2} + 50 \beta_1 - 266) q^{85} + (65 \beta_{4} - 14 \beta_{3} + 100 \beta_{2} + 4 \beta_1 - 506) q^{86} + ( - 193 \beta_{4} - 138 \beta_{3} - 64 \beta_{2} - 428 \beta_1 - 1470) q^{88} + ( - 152 \beta_{4} - 10 \beta_{3} - 100 \beta_{2} - 66 \beta_1 - 196) q^{89} + (94 \beta_{4} - 36 \beta_{3} - 14 \beta_{2} - 30 \beta_1 + 552) q^{91} + (22 \beta_{4} - 156 \beta_{3} - 14 \beta_1 - 716) q^{92} + (9 \beta_{4} - 118 \beta_{3} + 164 \beta_{2} - 38 \beta_1 - 1274) q^{94} + ( - 64 \beta_{4} + 68 \beta_{3} + 38 \beta_{2} - 2 \beta_1 - 476) q^{95} + ( - 32 \beta_{4} + 78 \beta_{3} + 48 \beta_{2} + 34 \beta_1 + 296) q^{97} + ( - 48 \beta_{4} + 16 \beta_{3} - 96 \beta_{2} - 67 \beta_1 + 704) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b4 + 2*b3 + 2*b1 + 6) * q^4 + (-2*b4 + b3 + b2 - b1 - 3) * q^5 + (-2*b4 - 2*b2 + 2*b1 + 8) * q^7 + (4*b4 + 4*b3 + 8*b2 + 9*b1 + 16) * q^8 + (6*b4 + 4*b2 + b1 - 12) * q^10 + (-9*b4 - 5*b3 - 10*b2 - 7*b1 - 3) * q^11 + (-6*b4 + 5*b3 - 9*b2 - 3*b1 + 5) * q^13 + (4*b4 + 14*b1 + 40) * q^14 + (9*b4 + 18*b3 + 16*b2 + 30*b1 + 30) * q^16 + (-4*b3 - 10*b2 - 10) * q^17 + (4*b4 - 2*b3 + 8*b1 + 44) * q^19 + (9*b4 + 2*b3 - 8*b2 - 8*b1 + 6) * q^20 + (-9*b4 - 34*b3 - 20*b2 - 28*b1 - 22) * q^22 + (-24*b4 - 14*b3 + 2*b2 - 46) * q^23 + (-2*b4 - 21*b3 + 11*b2 - 17*b1 + 32) * q^25 + (10*b4 - 24*b3 + 20*b2 + 25*b1 - 20) * q^26 + (22*b4 + 28*b3 + 16*b2 + 48*b1 + 116) * q^28 + 29 * q^29 + (5*b4 + 7*b3 - 14*b2 - 19*b1 + 93) * q^31 + (32*b4 + 60*b3 + 8*b2 + 81*b1 + 152) * q^32 + (-18*b4 - 20*b3 - 16*b2 - 26*b1 + 36) * q^34 + (30*b4 + 12*b3 + 42*b2 - 14*b1 + 8) * q^35 + (-28*b4 - 22*b3 + 30*b2 + 22*b1 + 48) * q^37 + (-4*b4 + 16*b3 - 8*b2 + 48*b1 + 104) * q^38 + (-78*b4 - 32*b3 - 24*b2 - 19*b1 - 44) * q^40 + (-44*b4 - 22*b3 - 16*b2 - 18*b1 + 216) * q^41 + (-11*b4 + 25*b3 + 22*b2 - 29*b1 - 49) * q^43 + (-26*b4 - 56*b3 - 56*b2 - 149*b1 - 156) * q^44 + (22*b4 + 4*b3 - 56*b2 - 78*b1 + 148) * q^46 + (7*b4 + 41*b3 + 16*b2 - 75*b1 - 45) * q^47 + (4*b4 - 24*b3 - 32*b2 + 40*b1 - 47) * q^49 + (-44*b4 - 12*b3 - 84*b2 - 84*b1 - 168) * q^50 + (25*b4 + 50*b3 - 24*b2 - 52*b1 + 326) * q^52 + (-10*b4 - 23*b3 - 17*b2 - 75*b1 + 109) * q^53 + (101*b4 + 41*b3 + 112*b2 - 13*b1 + 113) * q^55 + (44*b4 + 128*b3 + 112*b2 + 190*b1 + 120) * q^56 + 29*b1 * q^58 + (16*b4 - 30*b3 + 6*b2 - 52*b1 - 90) * q^59 + (12*b4 + 60*b3 + 66*b2 - 20*b1 + 114) * q^61 + (-29*b4 - 66*b3 + 28*b2 + 78*b1 - 286) * q^62 + (73*b4 + 34*b3 + 112*b2 + 282*b1 + 510) * q^64 + (146*b4 + 15*b3 + 49*b2 - 51*b1 + 373) * q^65 + (-44*b3 - 140*b2 - 88*b1 + 280) * q^67 + (-46*b4 - 52*b3 - 78*b1 - 100) * q^68 + (-8*b4 + 56*b3 + 48*b2 - 2*b1 - 448) * q^70 + (-6*b4 + 122*b3 + 96*b2 + 58*b1 + 122) * q^71 + (-32*b4 - 38*b3 - 66*b2 - 106*b1 - 384) * q^73 + (64*b4 + 104*b3 - 88*b2 + 32*b1 + 448) * q^74 + (48*b4 + 96*b3 + 64*b2 + 204*b1 + 288) * q^76 + (-48*b4 - 176*b3 - 158*b2 - 144*b1 + 332) * q^77 + (-177*b4 - 63*b3 - 140*b2 - 139*b1 + 27) * q^79 + (-23*b4 - 102*b3 - 64*b2 - 68*b1 + 174) * q^80 + (10*b4 - 68*b3 - 88*b2 + 136*b1 + 44) * q^82 + (-60*b4 - 14*b3 + 42*b2 + 96*b1 - 146) * q^83 + (136*b4 + 66*b3 + 20*b2 + 50*b1 - 266) * q^85 + (65*b4 - 14*b3 + 100*b2 + 4*b1 - 506) * q^86 + (-193*b4 - 138*b3 - 64*b2 - 428*b1 - 1470) * q^88 + (-152*b4 - 10*b3 - 100*b2 - 66*b1 - 196) * q^89 + (94*b4 - 36*b3 - 14*b2 - 30*b1 + 552) * q^91 + (22*b4 - 156*b3 - 14*b1 - 716) * q^92 + (9*b4 - 118*b3 + 164*b2 - 38*b1 - 1274) * q^94 + (-64*b4 + 68*b3 + 38*b2 - 2*b1 - 476) * q^95 + (-32*b4 + 78*b3 + 48*b2 + 34*b1 + 296) * q^97 + (-48*b4 + 16*b3 - 96*b2 - 67*b1 + 704) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8}+O(q^{10})$$ 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8 $$5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8} - 64 q^{10} - 12 q^{11} + 14 q^{13} + 192 q^{14} + 146 q^{16} - 66 q^{17} + 214 q^{19} - 6 q^{20} - 98 q^{22} - 164 q^{23} + 207 q^{25} - 56 q^{26} + 540 q^{28} + 145 q^{29} + 420 q^{31} + 652 q^{32} + 204 q^{34} + 52 q^{35} + 378 q^{37} + 496 q^{38} - 80 q^{40} + 1158 q^{41} - 204 q^{43} - 784 q^{44} + 580 q^{46} - 248 q^{47} - 283 q^{49} - 908 q^{50} + 1482 q^{52} + 554 q^{53} + 546 q^{55} + 608 q^{56} - 440 q^{59} + 618 q^{61} - 1250 q^{62} + 2594 q^{64} + 1656 q^{65} + 1164 q^{67} - 356 q^{68} - 2184 q^{70} + 692 q^{71} - 1950 q^{73} + 1832 q^{74} + 1376 q^{76} + 1616 q^{77} + 272 q^{79} + 890 q^{80} + 92 q^{82} - 512 q^{83} - 1628 q^{85} - 2446 q^{86} - 6954 q^{88} - 866 q^{89} + 2580 q^{91} - 3468 q^{92} - 5942 q^{94} - 2244 q^{95} + 1562 q^{97} + 3408 q^{98}+O(q^{100})$$ 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8 - 64 * q^10 - 12 * q^11 + 14 * q^13 + 192 * q^14 + 146 * q^16 - 66 * q^17 + 214 * q^19 - 6 * q^20 - 98 * q^22 - 164 * q^23 + 207 * q^25 - 56 * q^26 + 540 * q^28 + 145 * q^29 + 420 * q^31 + 652 * q^32 + 204 * q^34 + 52 * q^35 + 378 * q^37 + 496 * q^38 - 80 * q^40 + 1158 * q^41 - 204 * q^43 - 784 * q^44 + 580 * q^46 - 248 * q^47 - 283 * q^49 - 908 * q^50 + 1482 * q^52 + 554 * q^53 + 546 * q^55 + 608 * q^56 - 440 * q^59 + 618 * q^61 - 1250 * q^62 + 2594 * q^64 + 1656 * q^65 + 1164 * q^67 - 356 * q^68 - 2184 * q^70 + 692 * q^71 - 1950 * q^73 + 1832 * q^74 + 1376 * q^76 + 1616 * q^77 + 272 * q^79 + 890 * q^80 + 92 * q^82 - 512 * q^83 - 1628 * q^85 - 2446 * q^86 - 6954 * q^88 - 866 * q^89 + 2580 * q^91 - 3468 * q^92 - 5942 * q^94 - 2244 * q^95 + 1562 * q^97 + 3408 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + \nu^{2} - 8\nu - 2 ) / 2$$ (v^3 + v^2 - 8*v - 2) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} - \nu^{2} + 12\nu + 2 ) / 2$$ (-v^3 - v^2 + 12*v + 2) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 6$$ v^2 - 6 $$\beta_{4}$$ $$=$$ $$( \nu^{4} - 13\nu^{2} + 6\nu + 14 ) / 2$$ (v^4 - 13*v^2 + 6*v + 14) / 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 6$$ b3 + 6 $$\nu^{3}$$ $$=$$ $$-\beta_{3} + 4\beta_{2} + 6\beta _1 - 4$$ -b3 + 4*b2 + 6*b1 - 4 $$\nu^{4}$$ $$=$$ $$2\beta_{4} + 13\beta_{3} - 3\beta_{2} - 3\beta _1 + 64$$ 2*b4 + 13*b3 - 3*b2 - 3*b1 + 64

## 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
 −3.68360 0.328194 2.27399 −0.957567 3.03898
−4.47236 0 12.0020 6.52855 0 5.22706 −17.8986 0 −29.1981
1.2 −2.24125 0 −2.97681 −18.3339 0 −16.8583 24.6017 0 41.0908
1.3 −1.63099 0 −5.33986 16.8209 0 5.21997 21.7572 0 −27.4348
1.4 2.84972 0 0.120922 −12.8729 0 26.0540 −22.4532 0 −36.6841
1.5 5.49488 0 22.1937 −2.14270 0 20.3573 77.9928 0 −11.7739
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$29$$ $$-1$$

## 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 261.4.a.f 5
3.b odd 2 1 29.4.a.b 5
12.b even 2 1 464.4.a.l 5
15.d odd 2 1 725.4.a.c 5
21.c even 2 1 1421.4.a.e 5
24.f even 2 1 1856.4.a.bb 5
24.h odd 2 1 1856.4.a.y 5
87.d odd 2 1 841.4.a.b 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.4.a.b 5 3.b odd 2 1
261.4.a.f 5 1.a even 1 1 trivial
464.4.a.l 5 12.b even 2 1
725.4.a.c 5 15.d odd 2 1
841.4.a.b 5 87.d odd 2 1
1421.4.a.e 5 21.c even 2 1
1856.4.a.y 5 24.h odd 2 1
1856.4.a.bb 5 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} - 33T_{2}^{3} - 28T_{2}^{2} + 192T_{2} + 256$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(261))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 33 T^{3} - 28 T^{2} + \cdots + 256$$
$3$ $$T^{5}$$
$5$ $$T^{5} + 10 T^{4} - 366 T^{3} + \cdots + 55534$$
$7$ $$T^{5} - 40 T^{4} + 84 T^{3} + \cdots + 243968$$
$11$ $$T^{5} + 12 T^{4} - 4892 T^{3} + \cdots - 30997958$$
$13$ $$T^{5} - 14 T^{4} - 7558 T^{3} + \cdots - 13078418$$
$17$ $$T^{5} + 66 T^{4} - 2444 T^{3} + \cdots - 19935872$$
$19$ $$T^{5} - 214 T^{4} + \cdots + 19441152$$
$23$ $$T^{5} + 164 T^{4} + \cdots + 7938109184$$
$29$ $$(T - 29)^{5}$$
$31$ $$T^{5} - 420 T^{4} + 45552 T^{3} + \cdots - 2094346$$
$37$ $$T^{5} - 378 T^{4} + \cdots + 23564115968$$
$41$ $$T^{5} - 1158 T^{4} + \cdots - 59613728000$$
$43$ $$T^{5} + 204 T^{4} + \cdots + 198643410886$$
$47$ $$T^{5} + 248 T^{4} + \cdots + 203435244846$$
$53$ $$T^{5} - 554 T^{4} + \cdots + 786854101018$$
$59$ $$T^{5} + 440 T^{4} + \cdots - 109032704000$$
$61$ $$T^{5} - 618 T^{4} + \cdots + 2140697762176$$
$67$ $$T^{5} - 1164 T^{4} + \cdots - 39308070146048$$
$71$ $$T^{5} - 692 T^{4} + \cdots - 98341318953856$$
$73$ $$T^{5} + 1950 T^{4} + \cdots + 7201878016$$
$79$ $$T^{5} + \cdots - 240961986300538$$
$83$ $$T^{5} + 512 T^{4} + \cdots - 6057622580224$$
$89$ $$T^{5} + 866 T^{4} + \cdots - 21549994365568$$
$97$ $$T^{5} - 1562 T^{4} + \cdots - 20480102175488$$