# Properties

 Label 63.6.e.e Level $63$ Weight $6$ Character orbit 63.e Analytic conductor $10.104$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,6,Mod(37,63)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("63.37");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 63.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$10.1041806482$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436$$ x^8 - x^7 + 98*x^6 + 83*x^5 + 9122*x^4 - 91*x^3 + 28567*x^2 + 2058*x + 86436 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1 + 1) q^{2} + (\beta_{6} + \beta_{3} + 18 \beta_{2} - \beta_1) q^{4} + ( - \beta_{5} + \beta_{4} + \cdots - 2 \beta_1) q^{5}+ \cdots + (2 \beta_{7} + 27 \beta_{6} + \cdots - 37) q^{8}+O(q^{10})$$ q + (b2 - b1 + 1) * q^2 + (b6 + b3 + 18*b2 - b1) * q^4 + (-b5 + b4 + b3 + b2 - 2*b1) * q^5 + (b7 + 6*b6 + b5 + b4 + 45*b2 - 2*b1 + 53) * q^7 + (2*b7 + 27*b6 - 2*b5 - b4 - 37) * q^8 $$q + (\beta_{2} - \beta_1 + 1) q^{2} + (\beta_{6} + \beta_{3} + 18 \beta_{2} - \beta_1) q^{4} + ( - \beta_{5} + \beta_{4} + \cdots - 2 \beta_1) q^{5}+ \cdots + (88 \beta_{7} + 8219 \beta_{6} + \cdots - 86434) q^{98}+O(q^{100})$$ q + (b2 - b1 + 1) * q^2 + (b6 + b3 + 18*b2 - b1) * q^4 + (-b5 + b4 + b3 + b2 - 2*b1) * q^5 + (b7 + 6*b6 + b5 + b4 + 45*b2 - 2*b1 + 53) * q^7 + (2*b7 + 27*b6 - 2*b5 - b4 - 37) * q^8 + (23*b6 - b3 + 76*b2 - 23*b1) * q^10 + (3*b7 - 8*b6 - 9*b3 - 107*b2 + 8*b1) * q^11 + (9*b7 + 76*b6 - 9*b5 - b4 + 97) * q^13 + (4*b7 + 83*b6 - 9*b4 - b3 - 113*b2 - 114*b1 - 299) * q^14 + (-6*b5 - b4 - b3 - 840*b2 + 85*b1 - 839) * q^16 + (4*b7 - 108*b6 + 8*b3 - 92*b2 + 108*b1) * q^17 + (-15*b5 + b4 + b3 - 72*b2 - 220*b1 - 73) * q^19 + (30*b7 - 29*b6 - 30*b5 + 9*b4 - 1177) * q^20 + (-12*b7 - 419*b6 + 12*b5 - b4 + 449) * q^22 + (-20*b5 + 8*b4 + 8*b3 + 1804*b2 - 300*b1 + 1796) * q^23 + (45*b7 - 80*b6 - 95*b3 + 636*b2 + 80*b1) * q^25 + (-20*b5 - 103*b4 - 103*b3 - 3865*b2 + 116*b1 - 3762) * q^26 + (-26*b7 + 50*b6 + 40*b5 - 95*b4 - b3 + 896*b2 + 485*b1 - 2855) * q^28 + (-5*b7 + 254*b6 + 5*b5 + 103*b4 - 250) * q^29 + (-24*b7 + 130*b6 + 94*b3 - 1523*b2 - 130*b1) * q^31 + (50*b7 - 131*b6 - 71*b3 - 3948*b2 + 131*b1) * q^32 + (24*b7 + 312*b6 - 24*b5 + 96*b4 + 5160) * q^34 + (-37*b7 + 208*b6 + 29*b5 - 71*b4 + 104*b3 - 1706*b2 + 24*b1 + 3240) * q^35 + (-9*b5 + 95*b4 + 95*b3 - 3508*b2 - 1028*b1 - 3603) * q^37 + (-28*b7 - 316*b6 + 175*b3 + 10321*b2 + 316*b1) * q^38 + (-42*b5 - 93*b4 - 93*b3 + 1932*b2 + 633*b1 + 2025) * q^40 + (-142*b7 + 328*b6 + 142*b5 + 62*b4 - 1190) * q^41 + (-33*b7 + 816*b6 + 33*b5 + 93*b4 + 7049) * q^43 + (118*b5 + 167*b4 + 167*b3 + 17864*b2 - 379*b1 + 17697) * q^44 + (-24*b7 + 1752*b6 + 240*b3 + 16008*b2 - 1752*b1) * q^46 + (28*b5 + 56*b4 + 56*b3 - 3818*b2 + 324*b1 - 3874) * q^47 + (11*b7 + 1576*b6 - 61*b5 + 239*b4 + 94*b3 + 5315*b2 - 152*b1 - 5892) * q^49 + (-100*b7 - 2404*b6 + 100*b5 - 55*b4 + 2454) * q^50 + (42*b7 - 6036*b6 - 144*b3 - 13450*b2 + 6036*b1) * q^52 + (-239*b7 + 1338*b6 - 13*b3 - 3095*b2 - 1338*b1) * q^53 + (-315*b7 + 506*b6 + 315*b5 - 335*b4 + 18322) * q^55 + (26*b7 + 129*b6 - 8*b5 - 4*b4 - 159*b3 - 22170*b2 + 3429*b1 - 9514) * q^56 + (216*b5 - 239*b4 - 239*b3 - 12154*b2 - 4171*b1 - 11915) * q^58 + (-71*b7 - 888*b6 - 163*b3 - 9011*b2 + 888*b1) * q^59 + (240*b5 + 52*b4 + 52*b3 + 1566*b2 - 796*b1 + 1514) * q^61 + (140*b7 + 1875*b6 - 140*b5 - 58*b4 - 4447) * q^62 + (150*b7 - 3189*b6 - 150*b5 - 51*b4 - 17549) * q^64 + (-246*b5 - 330*b4 - 330*b3 + 16136*b2 + 1660*b1 + 16466) * q^65 + (-465*b7 - 2764*b6 - 193*b3 - 2286*b2 + 2764*b1) * q^67 + (272*b5 - 128*b4 - 128*b3 - 13312*b2 - 5280*b1 - 13184) * q^68 + (192*b7 + 2406*b6 - 276*b5 - 97*b4 - 145*b3 - 7658*b2 - 809*b1 - 4783) * q^70 + (180*b7 - 3660*b6 - 180*b5 + 24*b4 - 21414) * q^71 + (93*b7 - 3056*b6 - 143*b3 + 14074*b2 + 3056*b1) * q^73 + (172*b7 + 216*b6 + 1001*b3 + 46915*b2 - 216*b1) * q^74 + (774*b7 + 9924*b6 - 774*b5 + 432*b4 + 2630) * q^76 + (196*b7 + 228*b6 - 367*b5 + 1063*b4 + 183*b3 - 16115*b2 - 3730*b1 + 1232) * q^77 + (-786*b5 + 96*b4 + 96*b3 + 11635*b2 - 2286*b1 + 11539) * q^79 + (690*b7 - 3607*b6 - 1047*b3 + 6920*b2 + 3607*b1) * q^80 + (408*b5 + 98*b4 + 98*b3 - 13322*b2 - 4112*b1 - 13420) * q^82 + (129*b7 + 6432*b6 - 129*b5 - 1005*b4 + 51006) * q^83 + (372*b7 - 2988*b6 - 372*b5 - 192*b4 + 9924) * q^85 + (252*b5 - 717*b4 - 717*b3 - 31705*b2 - 11582*b1 - 30988) * q^86 + (186*b7 + 13545*b6 + 765*b3 + 25668*b2 - 13545*b1) * q^88 + (-622*b5 + 1582*b4 + 1582*b3 - 20966*b2 + 9508*b1 - 22548) * q^89 + (230*b7 + 1948*b6 + 203*b5 + 431*b4 - 47*b3 - 73960*b2 + 9748*b1 - 25799) * q^91 + (1072*b7 + 15792*b6 - 1072*b5 - 1424*b4 - 42800) * q^92 + (168*b7 - 990*b6 - 240*b3 - 18798*b2 + 990*b1) * q^94 + (294*b7 + 3820*b6 - 1542*b3 + 55528*b2 - 3820*b1) * q^95 + (-669*b7 + 464*b6 + 669*b5 - 863*b4 - 46842) * q^97 + (88*b7 + 8219*b6 + 268*b5 - 1609*b4 - 1607*b3 - 76298*b2 - 2987*b1 - 86434) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 3 q^{2} - 69 q^{4} + 258 q^{7} - 246 q^{8}+O(q^{10})$$ 8 * q + 3 * q^2 - 69 * q^4 + 258 * q^7 - 246 * q^8 $$8 q + 3 q^{2} - 69 q^{4} + 258 q^{7} - 246 q^{8} - 283 q^{10} + 402 q^{11} + 924 q^{13} - 1926 q^{14} - 3273 q^{16} + 276 q^{17} - 510 q^{19} - 9438 q^{20} + 2750 q^{22} + 6900 q^{23} - 2814 q^{25} - 15138 q^{26} - 26221 q^{28} - 1080 q^{29} + 6410 q^{31} + 15519 q^{32} + 42288 q^{34} + 33108 q^{35} - 15250 q^{37} - 41250 q^{38} + 8547 q^{40} - 8616 q^{41} + 58396 q^{43} + 70743 q^{44} - 61800 q^{46} - 15060 q^{47} - 64252 q^{49} + 14604 q^{50} + 47476 q^{52} + 13692 q^{53} + 146248 q^{55} + 15921 q^{56} - 52309 q^{58} + 34830 q^{59} + 5364 q^{61} - 32058 q^{62} - 146974 q^{64} + 66864 q^{65} + 5994 q^{67} - 58272 q^{68} - 4307 q^{70} - 178536 q^{71} - 59638 q^{73} - 185442 q^{74} + 42616 q^{76} + 75660 q^{77} + 44062 q^{79} - 33381 q^{80} - 57596 q^{82} + 416892 q^{83} + 72648 q^{85} - 136968 q^{86} - 87597 q^{88} - 77520 q^{89} + 104722 q^{91} - 316512 q^{92} + 73722 q^{94} - 221376 q^{95} - 377260 q^{97} - 382479 q^{98}+O(q^{100})$$ 8 * q + 3 * q^2 - 69 * q^4 + 258 * q^7 - 246 * q^8 - 283 * q^10 + 402 * q^11 + 924 * q^13 - 1926 * q^14 - 3273 * q^16 + 276 * q^17 - 510 * q^19 - 9438 * q^20 + 2750 * q^22 + 6900 * q^23 - 2814 * q^25 - 15138 * q^26 - 26221 * q^28 - 1080 * q^29 + 6410 * q^31 + 15519 * q^32 + 42288 * q^34 + 33108 * q^35 - 15250 * q^37 - 41250 * q^38 + 8547 * q^40 - 8616 * q^41 + 58396 * q^43 + 70743 * q^44 - 61800 * q^46 - 15060 * q^47 - 64252 * q^49 + 14604 * q^50 + 47476 * q^52 + 13692 * q^53 + 146248 * q^55 + 15921 * q^56 - 52309 * q^58 + 34830 * q^59 + 5364 * q^61 - 32058 * q^62 - 146974 * q^64 + 66864 * q^65 + 5994 * q^67 - 58272 * q^68 - 4307 * q^70 - 178536 * q^71 - 59638 * q^73 - 185442 * q^74 + 42616 * q^76 + 75660 * q^77 + 44062 * q^79 - 33381 * q^80 - 57596 * q^82 + 416892 * q^83 + 72648 * q^85 - 136968 * q^86 - 87597 * q^88 - 77520 * q^89 + 104722 * q^91 - 316512 * q^92 + 73722 * q^94 - 221376 * q^95 - 377260 * q^97 - 382479 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 8905 \nu^{7} + 366059 \nu^{6} - 1191820 \nu^{5} + 33153695 \nu^{4} - 47979268 \nu^{3} + \cdots + 307508418 ) / 9888988410$$ (-8905*v^7 + 366059*v^6 - 1191820*v^5 + 33153695*v^4 - 47979268*v^3 + 3262309295*v^2 - 141627885*v + 307508418) / 9888988410 $$\beta_{3}$$ $$=$$ $$( 1257 \nu^{7} - 279647 \nu^{6} + 388990 \nu^{5} - 26314019 \nu^{4} - 14452616 \nu^{3} + \cdots - 251390874 ) / 156968070$$ (1257*v^7 - 279647*v^6 + 388990*v^5 - 26314019*v^4 - 14452616*v^3 - 2382173465*v^2 - 51985031*v - 251390874) / 156968070 $$\beta_{4}$$ $$=$$ $$( 22795 \nu^{7} + 56368 \nu^{6} + 2163175 \nu^{5} + 2122285 \nu^{4} + 222588334 \nu^{3} + \cdots - 31645054854 ) / 706356315$$ (22795*v^7 + 56368*v^6 + 2163175*v^5 + 2122285*v^4 + 222588334*v^3 + 7196875*v^2 + 20796090*v - 31645054854) / 706356315 $$\beta_{5}$$ $$=$$ $$( - 341623 \nu^{7} - 8468248 \nu^{6} + 27571040 \nu^{5} - 966332554 \nu^{4} + \cdots - 235881275616 ) / 9888988410$$ (-341623*v^7 - 8468248*v^6 + 27571040*v^5 - 966332554*v^4 + 1109931296*v^3 - 75468829240*v^2 + 428233011339*v - 235881275616) / 9888988410 $$\beta_{6}$$ $$=$$ $$( - 25511 \nu^{7} + 22795 \nu^{6} - 2420915 \nu^{5} - 2375153 \nu^{4} - 232964210 \nu^{3} + \cdots - 54979470 ) / 706356315$$ (-25511*v^7 + 22795*v^6 - 2420915*v^5 - 2375153*v^4 - 232964210*v^3 - 8054375*v^2 - 23273922*v - 54979470) / 706356315 $$\beta_{7}$$ $$=$$ $$( 15589754 \nu^{7} - 23777815 \nu^{6} + 1539409445 \nu^{5} + 516927917 \nu^{4} + \cdots + 24613669710 ) / 9888988410$$ (15589754*v^7 - 23777815*v^6 + 1539409445*v^5 + 516927917*v^4 + 141447402185*v^3 - 70438948615*v^2 + 442767354393*v + 24613669710) / 9888988410
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{6} + \beta_{3} + 49\beta_{2} + \beta_1$$ -b6 + b3 + 49*b2 + b1 $$\nu^{3}$$ $$=$$ $$-2\beta_{7} - 91\beta_{6} + 2\beta_{5} - 2\beta_{4} - 44$$ -2*b7 - 91*b6 + 2*b5 - 2*b4 - 44 $$\nu^{4}$$ $$=$$ $$2\beta_{5} - 99\beta_{4} - 99\beta_{3} - 4505\beta_{2} - 181\beta _1 - 4406$$ 2*b5 - 99*b4 - 99*b3 - 4505*b2 - 181*b1 - 4406 $$\nu^{5}$$ $$=$$ $$196\beta_{7} + 8707\beta_{6} - 286\beta_{3} - 8624\beta_{2} - 8707\beta_1$$ 196*b7 + 8707*b6 - 286*b3 - 8624*b2 - 8707*b1 $$\nu^{6}$$ $$=$$ $$376\beta_{7} + 25333\beta_{6} - 376\beta_{5} + 9581\beta_{4} + 421300$$ 376*b7 + 25333*b6 - 376*b5 + 9581*b4 + 421300 $$\nu^{7}$$ $$=$$ $$-18786\beta_{5} + 36042\beta_{4} + 36042\beta_{3} + 1222350\beta_{2} + 841891\beta _1 + 1186308$$ -18786*b5 + 36042*b4 + 36042*b3 + 1222350*b2 + 841891*b1 + 1186308

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 5.09061 − 8.81720i 0.895402 − 1.55088i −0.874091 + 1.51397i −4.61193 + 7.98809i 5.09061 + 8.81720i 0.895402 + 1.55088i −0.874091 − 1.51397i −4.61193 − 7.98809i
−4.59061 + 7.95118i 0 −26.1475 45.2888i 11.0358 19.1146i 0 126.882 26.6059i 186.333 0 101.322 + 175.495i
37.2 −0.395402 + 0.684857i 0 15.6873 + 27.1712i −52.0958 + 90.2327i 0 −7.12980 129.446i −50.1170 0 −41.1977 71.3564i
37.3 1.37409 2.37999i 0 12.2237 + 21.1722i 29.1836 50.5475i 0 −21.4366 + 127.857i 155.128 0 −80.2019 138.914i
37.4 5.11193 8.85412i 0 −36.2636 62.8104i 11.8764 20.5705i 0 30.6840 125.958i −414.344 0 −121.423 210.310i
46.1 −4.59061 7.95118i 0 −26.1475 + 45.2888i 11.0358 + 19.1146i 0 126.882 + 26.6059i 186.333 0 101.322 175.495i
46.2 −0.395402 0.684857i 0 15.6873 27.1712i −52.0958 90.2327i 0 −7.12980 + 129.446i −50.1170 0 −41.1977 + 71.3564i
46.3 1.37409 + 2.37999i 0 12.2237 21.1722i 29.1836 + 50.5475i 0 −21.4366 127.857i 155.128 0 −80.2019 + 138.914i
46.4 5.11193 + 8.85412i 0 −36.2636 + 62.8104i 11.8764 + 20.5705i 0 30.6840 + 125.958i −414.344 0 −121.423 + 210.310i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.e.e 8
3.b odd 2 1 21.6.e.c 8
7.c even 3 1 inner 63.6.e.e 8
7.c even 3 1 441.6.a.w 4
7.d odd 6 1 441.6.a.v 4
12.b even 2 1 336.6.q.j 8
21.c even 2 1 147.6.e.o 8
21.g even 6 1 147.6.a.l 4
21.g even 6 1 147.6.e.o 8
21.h odd 6 1 21.6.e.c 8
21.h odd 6 1 147.6.a.m 4
84.n even 6 1 336.6.q.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.c 8 3.b odd 2 1
21.6.e.c 8 21.h odd 6 1
63.6.e.e 8 1.a even 1 1 trivial
63.6.e.e 8 7.c even 3 1 inner
147.6.a.l 4 21.g even 6 1
147.6.a.m 4 21.h odd 6 1
147.6.e.o 8 21.c even 2 1
147.6.e.o 8 21.g even 6 1
336.6.q.j 8 12.b even 2 1
336.6.q.j 8 84.n even 6 1
441.6.a.v 4 7.d odd 6 1
441.6.a.w 4 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 3T_{2}^{7} + 103T_{2}^{6} - 90T_{2}^{5} + 9190T_{2}^{4} - 16260T_{2}^{3} + 53772T_{2}^{2} + 37944T_{2} + 41616$$ acting on $$S_{6}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 3 T^{7} + \cdots + 41616$$
$3$ $$T^{8}$$
$5$ $$T^{8} + \cdots + 10164899803536$$
$7$ $$T^{8} + \cdots + 79\!\cdots\!01$$
$11$ $$T^{8} + \cdots + 28\!\cdots\!76$$
$13$ $$(T^{4} - 462 T^{3} + \cdots + 149501563456)^{2}$$
$17$ $$T^{8} + \cdots + 25\!\cdots\!24$$
$19$ $$T^{8} + \cdots + 54\!\cdots\!76$$
$23$ $$T^{8} + \cdots + 90\!\cdots\!76$$
$29$ $$(T^{4} + \cdots + 408027025117872)^{2}$$
$31$ $$T^{8} + \cdots + 75\!\cdots\!09$$
$37$ $$T^{8} + \cdots + 25\!\cdots\!36$$
$41$ $$(T^{4} + \cdots - 18\!\cdots\!52)^{2}$$
$43$ $$(T^{4} + \cdots - 991662745581932)^{2}$$
$47$ $$T^{8} + \cdots + 73\!\cdots\!36$$
$53$ $$T^{8} + \cdots + 72\!\cdots\!84$$
$59$ $$T^{8} + \cdots + 66\!\cdots\!96$$
$61$ $$T^{8} + \cdots + 32\!\cdots\!76$$
$67$ $$T^{8} + \cdots + 30\!\cdots\!56$$
$71$ $$(T^{4} + \cdots + 21\!\cdots\!68)^{2}$$
$73$ $$T^{8} + \cdots + 14\!\cdots\!00$$
$79$ $$T^{8} + \cdots + 27\!\cdots\!81$$
$83$ $$(T^{4} + \cdots - 41\!\cdots\!32)^{2}$$
$89$ $$T^{8} + \cdots + 22\!\cdots\!24$$
$97$ $$(T^{4} + \cdots - 11\!\cdots\!44)^{2}$$