# Properties

 Label 26.8.c.a Level $26$ Weight $8$ Character orbit 26.c Analytic conductor $8.122$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,8,Mod(3,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("26.3");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 26.c (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: $$8.12201066259$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 2341x^{4} - 26208x^{3} + 5480281x^{2} - 30676464x + 171714816$$ x^6 + 2341*x^4 - 26208*x^3 + 5480281*x^2 - 30676464*x + 171714816 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 8 \beta_{3} - 8) q^{2} - \beta_1 q^{3} + 64 \beta_{3} q^{4} + (3 \beta_{4} - 2 \beta_{2} - 110) q^{5} + (8 \beta_{2} + 8 \beta_1) q^{6} + (2 \beta_{5} - 386 \beta_{3} + \cdots + 17 \beta_1) q^{7}+ \cdots + ( - 11 \beta_{5} - 630 \beta_{3} + \cdots - 6 \beta_1) q^{9}+O(q^{10})$$ q + (-8*b3 - 8) * q^2 - b1 * q^3 + 64*b3 * q^4 + (3*b4 - 2*b2 - 110) * q^5 + (8*b2 + 8*b1) * q^6 + (2*b5 - 386*b3 + 17*b2 + 17*b1) * q^7 + 512 * q^8 + (-11*b5 - 630*b3 - 6*b2 - 6*b1) * q^9 $$q + ( - 8 \beta_{3} - 8) q^{2} - \beta_1 q^{3} + 64 \beta_{3} q^{4} + (3 \beta_{4} - 2 \beta_{2} - 110) q^{5} + (8 \beta_{2} + 8 \beta_1) q^{6} + (2 \beta_{5} - 386 \beta_{3} + \cdots + 17 \beta_1) q^{7}+ \cdots + ( - 26756 \beta_{4} + 19750 \beta_{2} - 527688) q^{99}+O(q^{100})$$ q + (-8*b3 - 8) * q^2 - b1 * q^3 + 64*b3 * q^4 + (3*b4 - 2*b2 - 110) * q^5 + (8*b2 + 8*b1) * q^6 + (2*b5 - 386*b3 + 17*b2 + 17*b1) * q^7 + 512 * q^8 + (-11*b5 - 630*b3 - 6*b2 - 6*b1) * q^9 + (-24*b5 - 24*b4 + 880*b3 - 16*b1 + 880) * q^10 + (-8*b5 - 8*b4 - 1412*b3 + 91*b1 - 1412) * q^11 - 64*b2 * q^12 + (-39*b5 - 52*b4 - 2392*b3 + 52*b2 + 182*b1 - 2457) * q^13 + (16*b4 - 136*b2 - 3088) * q^14 + (4*b5 + 4*b4 - 4140*b3 + 326*b1 - 4140) * q^15 + (-4096*b3 - 4096) * q^16 + (-248*b5 + 2163*b3 - 244*b2 - 244*b1) * q^17 + (-88*b4 + 48*b2 - 5040) * q^18 + (-168*b5 - 840*b3 + 299*b2 + 299*b1) * q^19 + (192*b5 - 7040*b3 + 128*b2 + 128*b1) * q^20 + (-175*b4 + 624*b2 + 27153) * q^21 + (64*b5 + 11296*b3 - 728*b2 - 728*b1) * q^22 + (586*b5 + 586*b4 + 12182*b3 + 657*b1 + 12182) * q^23 - 512*b1 * q^24 + (-29*b4 + 1226*b2 + 27575) * q^25 + (416*b5 + 104*b4 + 19656*b3 - 1456*b2 - 1040*b1 + 520) * q^26 + (2033*b2 - 13104) * q^27 + (-128*b5 - 128*b4 + 24704*b3 - 1088*b1 + 24704) * q^28 + (382*b5 + 382*b4 + 1391*b3 - 1110*b1 + 1391) * q^29 + (-32*b5 + 33120*b3 - 2608*b2 - 2608*b1) * q^30 + (1372*b4 + 96*b2 - 94664) * q^31 + 32768*b3 * q^32 + (1049*b5 - 138951*b3 + 1414*b2 + 1414*b1) * q^33 + (-1984*b4 + 1952*b2 + 17304) * q^34 + (-1020*b5 + 169720*b3 - 6550*b2 - 6550*b1) * q^35 + (704*b5 + 704*b4 + 40320*b3 + 384*b1 + 40320) * q^36 + (114*b5 + 114*b4 - 28617*b3 + 7810*b1 - 28617) * q^37 + (-1344*b4 - 2392*b2 - 6720) * q^38 + (1742*b5 - 494*b4 - 184626*b3 + 832*b2 - 299*b1 + 85410) * q^39 + (1536*b4 - 1024*b2 - 56320) * q^40 + (-4096*b5 - 4096*b4 + 256089*b3 - 3104*b1 + 256089) * q^41 + (1400*b5 + 1400*b4 - 217224*b3 + 4992*b1 - 217224) * q^42 + (-2200*b5 - 40736*b3 + 8915*b2 + 8915*b1) * q^43 + (512*b4 + 5824*b2 + 90368) * q^44 + (-2999*b5 - 268380*b3 + 1994*b2 + 1994*b1) * q^45 + (-4688*b5 - 97456*b3 - 5256*b2 - 5256*b1) * q^46 + (-652*b4 + 9172*b2 - 235636) * q^47 + (4096*b2 + 4096*b1) * q^48 + (4047*b5 + 4047*b4 + 164310*b3 + 19458*b1 + 164310) * q^49 + (232*b5 + 232*b4 - 220600*b3 + 9808*b1 - 220600) * q^50 + (1196*b4 - 20491*b2 - 464724) * q^51 + (-832*b5 + 2496*b4 - 4160*b3 + 8320*b2 - 3328*b1 + 153088) * q^52 + (-2861*b4 - 13370*b2 - 741846) * q^53 + (104832*b3 + 16264*b1 + 104832) * q^54 + (-5424*b5 - 5424*b4 + 304540*b3 - 31546*b1 + 304540) * q^55 + (1024*b5 - 197632*b3 + 8704*b2 + 8704*b1) * q^56 + (-4297*b4 - 8790*b2 + 408087) * q^57 + (-3056*b5 - 11128*b3 + 8880*b2 + 8880*b1) * q^58 + (9616*b5 - 937964*b3 - 22673*b2 - 22673*b1) * q^59 + (-256*b4 + 20864*b2 + 264960) * q^60 + (6746*b5 + 546145*b3 - 15238*b2 - 15238*b1) * q^61 + (-10976*b5 - 10976*b4 + 757312*b3 + 768*b1 + 757312) * q^62 + (-1440*b5 - 1440*b4 + 187236*b3 - 5618*b1 + 187236) * q^63 + 262144 * q^64 + (-11921*b5 - 13247*b4 - 92560*b3 - 18356*b2 - 59514*b1 - 670410) * q^65 + (8392*b4 - 11312*b2 - 1111608) * q^66 + (26480*b5 + 26480*b4 + 1345892*b3 - 14757*b1 + 1345892) * q^67 + (15872*b5 + 15872*b4 - 138432*b3 + 15616*b1 - 138432) * q^68 + (3711*b5 - 1223361*b3 + 31608*b2 + 31608*b1) * q^69 + (-8160*b4 + 52400*b2 + 1357760) * q^70 + (34722*b5 + 1129514*b3 + 6949*b2 + 6949*b1) * q^71 + (-5632*b5 - 322560*b3 - 3072*b2 - 3072*b1) * q^72 + (38499*b4 + 43974*b2 - 1311706) * q^73 + (-912*b5 + 228936*b3 - 62480*b2 - 62480*b1) * q^74 + (-13312*b5 - 13312*b4 + 1918800*b3 - 36903*b1 + 1918800) * q^75 + (10752*b5 + 10752*b4 + 53760*b3 - 19136*b1 + 53760) * q^76 + (17549*b4 - 71636*b2 - 2821411) * q^77 + (3952*b5 + 17888*b4 - 683280*b3 + 2392*b2 + 9048*b1 - 2160288) * q^78 + (-18720*b4 + 94212*b2 + 249908) * q^79 + (-12288*b5 - 12288*b4 + 450560*b3 - 8192*b1 + 450560) * q^80 + (-46420*b5 - 46420*b4 + 1787571*b3 - 12216*b1 + 1787571) * q^81 + (32768*b5 - 2048712*b3 + 24832*b2 + 24832*b1) * q^82 + (9156*b4 - 54048*b2 + 3701796) * q^83 + (-11200*b5 + 1737792*b3 - 39936*b2 - 39936*b1) * q^84 + (-18079*b5 - 8301210*b3 + 113134*b2 + 113134*b1) * q^85 + (-17600*b4 - 71320*b2 - 325888) * q^86 + (-14502*b5 + 1597626*b3 + 17925*b2 + 17925*b1) * q^87 + (-4096*b5 - 4096*b4 - 722944*b3 + 46592*b1 - 722944) * q^88 + (-30241*b5 - 30241*b4 + 860277*b3 + 179890*b1 + 860277) * q^89 + (-23992*b4 - 15952*b2 - 2147040) * q^90 + (10920*b5 + 30784*b4 - 1703000*b3 - 63401*b2 + 46215*b1 - 4916756) * q^91 + (-37504*b4 + 42048*b2 - 779648) * q^92 + (-9288*b5 - 9288*b4 - 319752*b3 + 187384*b1 - 319752) * q^93 + (5216*b5 + 5216*b4 + 1885088*b3 + 73376*b1 + 1885088) * q^94 + (-21020*b5 - 3447660*b3 - 79330*b2 - 79330*b1) * q^95 - 32768*b2 * q^96 + (-51203*b5 + 2999903*b3 + 49034*b2 + 49034*b1) * q^97 + (-32376*b5 - 1314480*b3 - 155664*b2 - 155664*b1) * q^98 + (-26756*b4 + 19750*b2 - 527688) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 24 q^{2} - 192 q^{4} - 666 q^{5} + 1160 q^{7} + 3072 q^{8} + 1879 q^{9}+O(q^{10})$$ 6 * q - 24 * q^2 - 192 * q^4 - 666 * q^5 + 1160 * q^7 + 3072 * q^8 + 1879 * q^9 $$6 q - 24 q^{2} - 192 q^{4} - 666 q^{5} + 1160 q^{7} + 3072 q^{8} + 1879 q^{9} + 2664 q^{10} - 4228 q^{11} - 7501 q^{13} - 18560 q^{14} - 12424 q^{15} - 12288 q^{16} - 6737 q^{17} - 30064 q^{18} + 2352 q^{19} + 21312 q^{20} + 163268 q^{21} - 33824 q^{22} + 35960 q^{23} + 165508 q^{25} - 55640 q^{26} - 78624 q^{27} + 74240 q^{28} + 3791 q^{29} - 99392 q^{30} - 570728 q^{31} - 98304 q^{32} + 417902 q^{33} + 107792 q^{34} - 510180 q^{35} + 120256 q^{36} - 85965 q^{37} - 37632 q^{38} + 1069068 q^{39} - 340992 q^{40} + 772363 q^{41} - 653072 q^{42} + 120008 q^{43} + 541184 q^{44} + 802141 q^{45} + 287680 q^{46} - 1412512 q^{47} + 488883 q^{49} - 662032 q^{50} - 2790736 q^{51} + 925184 q^{52} - 4445354 q^{53} + 314496 q^{54} + 919044 q^{55} + 593920 q^{56} + 2457116 q^{57} + 30328 q^{58} + 2823508 q^{59} + 1590272 q^{60} - 1631689 q^{61} + 2282912 q^{62} + 563148 q^{63} + 1572864 q^{64} - 3730207 q^{65} - 6686432 q^{66} + 4011196 q^{67} - 431168 q^{68} + 3673794 q^{69} + 8162880 q^{70} - 3353820 q^{71} + 962048 q^{72} - 7947234 q^{73} - 687720 q^{74} + 5769712 q^{75} + 150528 q^{76} - 16963564 q^{77} - 10943712 q^{78} + 1536888 q^{79} + 1363968 q^{80} + 5409133 q^{81} + 6178904 q^{82} + 22192464 q^{83} - 5224576 q^{84} + 24885551 q^{85} - 1920128 q^{86} - 4807380 q^{87} - 2164736 q^{88} + 2611072 q^{89} - 12834256 q^{90} - 24442184 q^{91} - 4602880 q^{92} - 949968 q^{93} + 5650048 q^{94} + 10321960 q^{95} - 9050912 q^{97} + 3911064 q^{98} - 3112616 q^{99}+O(q^{100})$$ 6 * q - 24 * q^2 - 192 * q^4 - 666 * q^5 + 1160 * q^7 + 3072 * q^8 + 1879 * q^9 + 2664 * q^10 - 4228 * q^11 - 7501 * q^13 - 18560 * q^14 - 12424 * q^15 - 12288 * q^16 - 6737 * q^17 - 30064 * q^18 + 2352 * q^19 + 21312 * q^20 + 163268 * q^21 - 33824 * q^22 + 35960 * q^23 + 165508 * q^25 - 55640 * q^26 - 78624 * q^27 + 74240 * q^28 + 3791 * q^29 - 99392 * q^30 - 570728 * q^31 - 98304 * q^32 + 417902 * q^33 + 107792 * q^34 - 510180 * q^35 + 120256 * q^36 - 85965 * q^37 - 37632 * q^38 + 1069068 * q^39 - 340992 * q^40 + 772363 * q^41 - 653072 * q^42 + 120008 * q^43 + 541184 * q^44 + 802141 * q^45 + 287680 * q^46 - 1412512 * q^47 + 488883 * q^49 - 662032 * q^50 - 2790736 * q^51 + 925184 * q^52 - 4445354 * q^53 + 314496 * q^54 + 919044 * q^55 + 593920 * q^56 + 2457116 * q^57 + 30328 * q^58 + 2823508 * q^59 + 1590272 * q^60 - 1631689 * q^61 + 2282912 * q^62 + 563148 * q^63 + 1572864 * q^64 - 3730207 * q^65 - 6686432 * q^66 + 4011196 * q^67 - 431168 * q^68 + 3673794 * q^69 + 8162880 * q^70 - 3353820 * q^71 + 962048 * q^72 - 7947234 * q^73 - 687720 * q^74 + 5769712 * q^75 + 150528 * q^76 - 16963564 * q^77 - 10943712 * q^78 + 1536888 * q^79 + 1363968 * q^80 + 5409133 * q^81 + 6178904 * q^82 + 22192464 * q^83 - 5224576 * q^84 + 24885551 * q^85 - 1920128 * q^86 - 4807380 * q^87 - 2164736 * q^88 + 2611072 * q^89 - 12834256 * q^90 - 24442184 * q^91 - 4602880 * q^92 - 949968 * q^93 + 5650048 * q^94 + 10321960 * q^95 - 9050912 * q^97 + 3911064 * q^98 - 3112616 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 2341x^{4} - 26208x^{3} + 5480281x^{2} - 30676464x + 171714816$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 13104 ) / 2341$$ (v^3 - 13104) / 2341 $$\beta_{3}$$ $$=$$ $$( \nu^{5} + 2341\nu^{3} - 13104\nu^{2} + 5480281\nu - 30676464 ) / 30676464$$ (v^5 + 2341*v^3 - 13104*v^2 + 5480281*v - 30676464) / 30676464 $$\beta_{4}$$ $$=$$ $$( \nu^{4} + 6\nu^{3} + 2341\nu^{2} - 13104\nu + 3566313 ) / 25751$$ (v^4 + 6*v^3 + 2341*v^2 - 13104*v + 3566313) / 25751 $$\beta_{5}$$ $$=$$ $$( 173\nu^{5} + 396257\nu^{3} - 5675488\nu^{2} + 927637637\nu - 5192551728 ) / 37493456$$ (173*v^5 + 396257*v^3 - 5675488*v^2 + 927637637*v - 5192551728) / 37493456
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-11\beta_{5} + 1557\beta_{3} - 6\beta_{2} - 6\beta_1$$ -11*b5 + 1557*b3 - 6*b2 - 6*b1 $$\nu^{3}$$ $$=$$ $$2341\beta_{2} + 13104$$ 2341*b2 + 13104 $$\nu^{4}$$ $$=$$ $$25751\beta_{5} + 25751\beta_{4} - 3644937\beta_{3} + 27150\beta _1 - 3644937$$ 25751*b5 + 25751*b4 - 3644937*b3 + 27150*b1 - 3644937 $$\nu^{5}$$ $$=$$ $$-144144\beta_{5} + 51079392\beta_{3} - 5558905\beta_{2} - 5558905\beta_1$$ -144144*b5 + 51079392*b3 - 5558905*b2 - 5558905*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/26\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$\chi(n)$$ $$\beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 22.6479 + 39.2272i 2.83785 + 4.91531i −25.4857 − 44.1425i 22.6479 − 39.2272i 2.83785 − 4.91531i −25.4857 + 44.1425i
−4.00000 6.92820i −22.6479 39.2272i −32.0000 + 55.4256i −228.447 −181.183 + 313.818i −122.334 + 211.889i 512.000 67.6495 117.172i 913.789 + 1582.73i
3.2 −4.00000 6.92820i −2.83785 4.91531i −32.0000 + 55.4256i 307.915 −22.7028 + 39.3225i 9.23537 15.9961i 512.000 1077.39 1866.10i −1231.66 2133.30i
3.3 −4.00000 6.92820i 25.4857 + 44.1425i −32.0000 + 55.4256i −412.467 203.886 353.140i 693.099 1200.48i 512.000 −205.543 + 356.010i 1649.87 + 2857.66i
9.1 −4.00000 + 6.92820i −22.6479 + 39.2272i −32.0000 55.4256i −228.447 −181.183 313.818i −122.334 211.889i 512.000 67.6495 + 117.172i 913.789 1582.73i
9.2 −4.00000 + 6.92820i −2.83785 + 4.91531i −32.0000 55.4256i 307.915 −22.7028 39.3225i 9.23537 + 15.9961i 512.000 1077.39 + 1866.10i −1231.66 + 2133.30i
9.3 −4.00000 + 6.92820i 25.4857 44.1425i −32.0000 55.4256i −412.467 203.886 + 353.140i 693.099 + 1200.48i 512.000 −205.543 356.010i 1649.87 2857.66i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.3 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.8.c.a 6
3.b odd 2 1 234.8.h.a 6
4.b odd 2 1 208.8.i.a 6
13.c even 3 1 inner 26.8.c.a 6
13.c even 3 1 338.8.a.h 3
13.e even 6 1 338.8.a.g 3
13.f odd 12 2 338.8.b.g 6
39.i odd 6 1 234.8.h.a 6
52.j odd 6 1 208.8.i.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.c.a 6 1.a even 1 1 trivial
26.8.c.a 6 13.c even 3 1 inner
208.8.i.a 6 4.b odd 2 1
208.8.i.a 6 52.j odd 6 1
234.8.h.a 6 3.b odd 2 1
234.8.h.a 6 39.i odd 6 1
338.8.a.g 3 13.e even 6 1
338.8.a.h 3 13.c even 3 1
338.8.b.g 6 13.f odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 2341T_{3}^{4} + 26208T_{3}^{3} + 5480281T_{3}^{2} + 30676464T_{3} + 171714816$$ acting on $$S_{8}^{\mathrm{new}}(26, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 8 T + 64)^{3}$$
$3$ $$T^{6} + 2341 T^{4} + \cdots + 171714816$$
$5$ $$(T^{3} + 333 T^{2} + \cdots - 29013900)^{2}$$
$7$ $$T^{6} + \cdots + 39243960250000$$
$11$ $$T^{6} + \cdots + 12\!\cdots\!76$$
$13$ $$T^{6} + \cdots + 24\!\cdots\!13$$
$17$ $$T^{6} + \cdots + 73\!\cdots\!41$$
$19$ $$T^{6} + \cdots + 67\!\cdots\!00$$
$23$ $$T^{6} + \cdots + 40\!\cdots\!64$$
$29$ $$T^{6} + \cdots + 13\!\cdots\!69$$
$31$ $$(T^{3} + \cdots - 32\!\cdots\!72)^{2}$$
$37$ $$T^{6} + \cdots + 18\!\cdots\!81$$
$41$ $$T^{6} + \cdots + 76\!\cdots\!25$$
$43$ $$T^{6} + \cdots + 26\!\cdots\!36$$
$47$ $$(T^{3} + \cdots - 62\!\cdots\!68)^{2}$$
$53$ $$(T^{3} + \cdots - 81\!\cdots\!56)^{2}$$
$59$ $$T^{6} + \cdots + 70\!\cdots\!04$$
$61$ $$T^{6} + \cdots + 54\!\cdots\!25$$
$67$ $$T^{6} + \cdots + 34\!\cdots\!44$$
$71$ $$T^{6} + \cdots + 78\!\cdots\!16$$
$73$ $$(T^{3} + \cdots - 35\!\cdots\!60)^{2}$$
$79$ $$(T^{3} + \cdots - 48\!\cdots\!72)^{2}$$
$83$ $$(T^{3} + \cdots - 96\!\cdots\!08)^{2}$$
$89$ $$T^{6} + \cdots + 24\!\cdots\!76$$
$97$ $$T^{6} + \cdots + 15\!\cdots\!44$$