# Properties

 Label 10.7.c.b Level 10 Weight 7 Character orbit 10.c Analytic conductor 2.301 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$10 = 2 \cdot 5$$ Weight: $$k$$ = $$7$$ Character orbit: $$[\chi]$$ = 10.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.30054083620$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{129})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 4 + 4 \beta_{1} ) q^{2} + ( -5 + 4 \beta_{1} - \beta_{3} ) q^{3} + 32 \beta_{1} q^{4} + ( 83 + 28 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{5} + ( -40 - 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{6} + ( -45 - 45 \beta_{1} - 11 \beta_{2} ) q^{7} + ( -128 + 128 \beta_{1} ) q^{8} + ( -924 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 4 + 4 \beta_{1} ) q^{2} + ( -5 + 4 \beta_{1} - \beta_{3} ) q^{3} + 32 \beta_{1} q^{4} + ( 83 + 28 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{5} + ( -40 - 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{6} + ( -45 - 45 \beta_{1} - 11 \beta_{2} ) q^{7} + ( -128 + 128 \beta_{1} ) q^{8} + ( -924 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} ) q^{9} + ( 228 + 448 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} ) q^{10} + ( -570 + 13 \beta_{1} - 13 \beta_{2} + 13 \beta_{3} ) q^{11} + ( -160 - 160 \beta_{1} + 32 \beta_{2} ) q^{12} + ( 197 - 199 \beta_{1} - 2 \beta_{3} ) q^{13} + ( -404 \beta_{1} - 44 \beta_{2} - 44 \beta_{3} ) q^{14} + ( 1067 + 3422 \beta_{1} + 14 \beta_{2} - 87 \beta_{3} ) q^{15} -1024 q^{16} + ( 3069 + 3069 \beta_{1} + 46 \beta_{2} ) q^{17} + ( 3732 - 3660 \beta_{1} + 72 \beta_{3} ) q^{18} + ( -2750 \beta_{1} + 34 \beta_{2} + 34 \beta_{3} ) q^{19} + ( -832 + 2688 \beta_{1} - 64 \beta_{2} + 32 \beta_{3} ) q^{20} + ( -17282 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{21} + ( -2280 - 2280 \beta_{1} - 104 \beta_{2} ) q^{22} + ( 8795 - 8876 \beta_{1} - 81 \beta_{3} ) q^{23} + ( -1152 \beta_{1} + 128 \beta_{2} + 128 \beta_{3} ) q^{24} + ( -235 - 135 \beta_{1} + 55 \beta_{2} + 385 \beta_{3} ) q^{25} + ( 1576 - 8 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{26} + ( 15528 + 15528 \beta_{1} - 276 \beta_{2} ) q^{27} + ( 1440 - 1792 \beta_{1} - 352 \beta_{3} ) q^{28} + ( -14320 \beta_{1} - 232 \beta_{2} - 232 \beta_{3} ) q^{29} + ( -9768 + 18012 \beta_{1} + 404 \beta_{2} - 292 \beta_{3} ) q^{30} + ( -5082 - 349 \beta_{1} + 349 \beta_{2} - 349 \beta_{3} ) q^{31} + ( -4096 - 4096 \beta_{1} ) q^{32} + ( -18106 + 18572 \beta_{1} + 466 \beta_{3} ) q^{33} + ( 24736 \beta_{1} + 184 \beta_{2} + 184 \beta_{3} ) q^{34} + ( 32899 - 23091 \beta_{1} - 857 \beta_{2} - 454 \beta_{3} ) q^{35} + ( 29856 + 288 \beta_{1} - 288 \beta_{2} + 288 \beta_{3} ) q^{36} + ( -17631 - 17631 \beta_{1} + 1356 \beta_{2} ) q^{37} + ( 11136 - 10864 \beta_{1} + 272 \beta_{3} ) q^{38} + ( -1443 \beta_{1} - 189 \beta_{2} - 189 \beta_{3} ) q^{39} + ( -13952 + 7168 \beta_{1} - 384 \beta_{2} - 128 \beta_{3} ) q^{40} + ( 28846 + 1289 \beta_{1} - 1289 \beta_{2} + 1289 \beta_{3} ) q^{41} + ( -69128 - 69128 \beta_{1} - 8 \beta_{2} ) q^{42} + ( -73301 + 73468 \beta_{1} + 167 \beta_{3} ) q^{43} + ( -18656 \beta_{1} - 416 \beta_{2} - 416 \beta_{3} ) q^{44} + ( -19266 - 91881 \beta_{1} + 2343 \beta_{2} + 66 \beta_{3} ) q^{45} + ( 70360 - 324 \beta_{1} + 324 \beta_{2} - 324 \beta_{3} ) q^{46} + ( 49755 + 49755 \beta_{1} - 1791 \beta_{2} ) q^{47} + ( 5120 - 4096 \beta_{1} + 1024 \beta_{3} ) q^{48} + ( 82564 \beta_{1} + 1111 \beta_{2} + 1111 \beta_{3} ) q^{49} + ( 1140 - 1260 \beta_{1} - 1320 \beta_{2} + 1760 \beta_{3} ) q^{50} + ( 43462 - 2885 \beta_{1} + 2885 \beta_{2} - 2885 \beta_{3} ) q^{51} + ( 6304 + 6304 \beta_{1} + 64 \beta_{2} ) q^{52} + ( -32903 + 28091 \beta_{1} - 4812 \beta_{3} ) q^{53} + ( 123120 \beta_{1} - 1104 \beta_{2} - 1104 \beta_{3} ) q^{54} + ( -26354 - 78139 \beta_{1} - 2013 \beta_{2} - 451 \beta_{3} ) q^{55} + ( 11520 - 1408 \beta_{1} + 1408 \beta_{2} - 1408 \beta_{3} ) q^{56} + ( 68728 + 68728 \beta_{1} - 3056 \beta_{2} ) q^{57} + ( 56352 - 58208 \beta_{1} - 1856 \beta_{3} ) q^{58} + ( -82390 \beta_{1} + 2586 \beta_{2} + 2586 \beta_{3} ) q^{59} + ( -112288 + 34592 \beta_{1} + 2784 \beta_{2} + 448 \beta_{3} ) q^{60} + ( -56266 + 4577 \beta_{1} - 4577 \beta_{2} + 4577 \beta_{3} ) q^{61} + ( -20328 - 20328 \beta_{1} + 2792 \beta_{2} ) q^{62} + ( 117603 - 108348 \beta_{1} + 9255 \beta_{3} ) q^{63} -32768 \beta_{1} q^{64} + ( 24697 - 4748 \beta_{1} + 649 \beta_{2} + 33 \beta_{3} ) q^{65} + ( -144848 + 1864 \beta_{1} - 1864 \beta_{2} + 1864 \beta_{3} ) q^{66} + ( -196189 - 196189 \beta_{1} + 5577 \beta_{2} ) q^{67} + ( -98208 + 99680 \beta_{1} + 1472 \beta_{3} ) q^{68} + ( -51093 \beta_{1} - 8471 \beta_{2} - 8471 \beta_{3} ) q^{69} + ( 222144 + 35804 \beta_{1} - 1612 \beta_{2} - 5244 \beta_{3} ) q^{70} + ( -261706 - 173 \beta_{1} + 173 \beta_{2} - 173 \beta_{3} ) q^{71} + ( 119424 + 119424 \beta_{1} - 2304 \beta_{2} ) q^{72} + ( 228897 - 232289 \beta_{1} - 3392 \beta_{3} ) q^{73} + ( -135624 \beta_{1} + 5424 \beta_{2} + 5424 \beta_{3} ) q^{74} + ( 92435 + 620960 \beta_{1} - 2280 \beta_{2} - 1085 \beta_{3} ) q^{75} + ( 89088 + 1088 \beta_{1} - 1088 \beta_{2} + 1088 \beta_{3} ) q^{76} + ( 256166 + 256166 \beta_{1} + 7726 \beta_{2} ) q^{77} + ( 5016 - 6528 \beta_{1} - 1512 \beta_{3} ) q^{78} + ( -179280 \beta_{1} + 2704 \beta_{2} + 2704 \beta_{3} ) q^{79} + ( -84992 - 28672 \beta_{1} - 1024 \beta_{2} - 2048 \beta_{3} ) q^{80} + ( 79965 - 10071 \beta_{1} + 10071 \beta_{2} - 10071 \beta_{3} ) q^{81} + ( 115384 + 115384 \beta_{1} - 10312 \beta_{2} ) q^{82} + ( -312797 + 306232 \beta_{1} - 6565 \beta_{3} ) q^{83} + ( -553056 \beta_{1} - 32 \beta_{2} - 32 \beta_{3} ) q^{84} + ( 26629 + 419214 \beta_{1} + 703 \beta_{2} + 10541 \beta_{3} ) q^{85} + ( -586408 + 668 \beta_{1} - 668 \beta_{2} + 668 \beta_{3} ) q^{86} + ( -303544 - 303544 \beta_{1} - 12232 \beta_{2} ) q^{87} + ( 72960 - 76288 \beta_{1} - 3328 \beta_{3} ) q^{88} + ( 89680 \beta_{1} + 1920 \beta_{2} + 1920 \beta_{3} ) q^{89} + ( 290724 - 435216 \beta_{1} + 9108 \beta_{2} + 9636 \beta_{3} ) q^{90} + ( -53194 + 2279 \beta_{1} - 2279 \beta_{2} + 2279 \beta_{3} ) q^{91} + ( 281440 + 281440 \beta_{1} + 2592 \beta_{2} ) q^{92} + ( 587998 - 580124 \beta_{1} + 7874 \beta_{3} ) q^{93} + ( 390876 \beta_{1} - 7164 \beta_{2} - 7164 \beta_{3} ) q^{94} + ( -92040 - 284890 \beta_{1} + 7370 \beta_{2} + 990 \beta_{3} ) q^{95} + ( 40960 + 4096 \beta_{1} - 4096 \beta_{2} + 4096 \beta_{3} ) q^{96} + ( -479903 - 479903 \beta_{1} - 8960 \beta_{2} ) q^{97} + ( -325812 + 334700 \beta_{1} + 8888 \beta_{3} ) q^{98} + ( 161367 \beta_{1} + 6765 \beta_{2} + 6765 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 16q^{2} - 18q^{3} + 330q^{5} - 144q^{6} - 202q^{7} - 512q^{8} + O(q^{10})$$ $$4q + 16q^{2} - 18q^{3} + 330q^{5} - 144q^{6} - 202q^{7} - 512q^{8} + 880q^{10} - 2332q^{11} - 576q^{12} + 792q^{13} + 4470q^{15} - 4096q^{16} + 12368q^{17} + 14784q^{18} - 3520q^{20} - 69132q^{21} - 9328q^{22} + 35342q^{23} - 1600q^{25} + 6336q^{26} + 61560q^{27} + 6464q^{28} - 37680q^{30} - 18932q^{31} - 16384q^{32} - 73356q^{33} + 130790q^{35} + 118272q^{36} - 67812q^{37} + 44000q^{38} - 56320q^{40} + 110228q^{41} - 276528q^{42} - 293538q^{43} - 72510q^{45} + 282736q^{46} + 195438q^{47} + 18432q^{48} - 1600q^{50} + 185388q^{51} + 25344q^{52} - 121988q^{53} - 108540q^{55} + 51712q^{56} + 268800q^{57} + 229120q^{58} - 444480q^{60} - 243372q^{61} - 75728q^{62} + 451902q^{63} + 100020q^{65} - 586848q^{66} - 773602q^{67} - 395776q^{68} + 895840q^{70} - 1046132q^{71} + 473088q^{72} + 922372q^{73} + 367350q^{75} + 352000q^{76} + 1040116q^{77} + 23088q^{78} - 337920q^{80} + 360144q^{81} + 440912q^{82} - 1238058q^{83} + 86840q^{85} - 2348304q^{86} - 1238640q^{87} + 298496q^{88} + 1161840q^{90} - 221892q^{91} + 1130944q^{92} + 2336244q^{93} - 355400q^{95} + 147456q^{96} - 1937532q^{97} - 1321024q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 65 x^{2} + 1024$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 33 \nu$$$$)/32$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{3} + 160 \nu^{2} + 259 \nu + 5216$$$$)/32$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 80 \nu^{2} + 113 \nu - 2608$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 5 \beta_{1}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} - \beta_{1} - 326$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-33 \beta_{3} - 33 \beta_{2} + 485 \beta_{1}$$$$)/10$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$\chi(n)$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 6.17891i − 5.17891i − 6.17891i 5.17891i
4.00000 4.00000i −32.8945 32.8945i 32.0000i 110.895 + 57.6836i −263.156 261.840 261.840i −128.000 128.000i 1435.10i 674.313 212.844i
3.2 4.00000 4.00000i 23.8945 + 23.8945i 32.0000i 54.1055 112.684i 191.156 −362.840 + 362.840i −128.000 128.000i 412.898i −234.313 667.156i
7.1 4.00000 + 4.00000i −32.8945 + 32.8945i 32.0000i 110.895 57.6836i −263.156 261.840 + 261.840i −128.000 + 128.000i 1435.10i 674.313 + 212.844i
7.2 4.00000 + 4.00000i 23.8945 23.8945i 32.0000i 54.1055 + 112.684i 191.156 −362.840 362.840i −128.000 + 128.000i 412.898i −234.313 + 667.156i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.7.c.b 4
3.b odd 2 1 90.7.g.b 4
4.b odd 2 1 80.7.p.c 4
5.b even 2 1 50.7.c.d 4
5.c odd 4 1 inner 10.7.c.b 4
5.c odd 4 1 50.7.c.d 4
15.d odd 2 1 450.7.g.m 4
15.e even 4 1 90.7.g.b 4
15.e even 4 1 450.7.g.m 4
20.e even 4 1 80.7.p.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.7.c.b 4 1.a even 1 1 trivial
10.7.c.b 4 5.c odd 4 1 inner
50.7.c.d 4 5.b even 2 1
50.7.c.d 4 5.c odd 4 1
80.7.p.c 4 4.b odd 2 1
80.7.p.c 4 20.e even 4 1
90.7.g.b 4 3.b odd 2 1
90.7.g.b 4 15.e even 4 1
450.7.g.m 4 15.d odd 2 1
450.7.g.m 4 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 18 T_{3}^{3} + 162 T_{3}^{2} - 28296 T_{3} + 2471184$$ acting on $$S_{7}^{\mathrm{new}}(10, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 8 T + 32 T^{2} )^{2}$$
$3$ $$1 + 18 T + 162 T^{2} - 15174 T^{3} - 1049886 T^{4} - 11061846 T^{5} + 86093442 T^{6} + 6973568802 T^{7} + 282429536481 T^{8}$$
$5$ $$1 - 330 T + 55250 T^{2} - 5156250 T^{3} + 244140625 T^{4}$$
$7$ $$1 + 202 T + 20402 T^{2} - 14617326 T^{3} - 25631752606 T^{4} - 1719713786574 T^{5} + 282389941474802 T^{6} + 328939546777910698 T^{7} +$$$$19\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 + 1166 T + 3337986 T^{2} + 2065640126 T^{3} + 3138428376721 T^{4} )^{2}$$
$13$ $$1 - 792 T + 313632 T^{2} - 3879823464 T^{3} + 47990658286814 T^{4} - 18727166814446376 T^{5} + 7307025033133960992 T^{6} -$$$$89\!\cdots\!68$$$$T^{7} +$$$$54\!\cdots\!61$$$$T^{8}$$
$17$ $$1 - 12368 T + 76483712 T^{2} - 492820856496 T^{3} + 2928709829028734 T^{4} - 11895497428311298224 T^{5} +$$$$44\!\cdots\!32$$$$T^{6} -$$$$17\!\cdots\!12$$$$T^{7} +$$$$33\!\cdots\!21$$$$T^{8}$$
$19$ $$1 - 165602324 T^{2} + 11169887241682566 T^{4} -$$$$36\!\cdots\!64$$$$T^{6} +$$$$48\!\cdots\!21$$$$T^{8}$$
$23$ $$1 - 35342 T + 624528482 T^{2} - 10376001126774 T^{3} + 151202761120969154 T^{4} -$$$$15\!\cdots\!86$$$$T^{5} +$$$$13\!\cdots\!22$$$$T^{6} -$$$$11\!\cdots\!98$$$$T^{7} +$$$$48\!\cdots\!41$$$$T^{8}$$
$29$ $$1 - 1622003684 T^{2} + 1222972659971291046 T^{4} -$$$$57\!\cdots\!44$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$
$31$ $$( 1 + 9466 T + 1404600426 T^{2} + 8401109844346 T^{3} + 787662783788549761 T^{4} )^{2}$$
$37$ $$1 + 67812 T + 2299233672 T^{2} + 11905958106924 T^{3} - 5651181413866152466 T^{4} +$$$$30\!\cdots\!16$$$$T^{5} +$$$$15\!\cdots\!32$$$$T^{6} +$$$$11\!\cdots\!48$$$$T^{7} +$$$$43\!\cdots\!61$$$$T^{8}$$
$41$ $$( 1 - 55114 T + 4901191506 T^{2} - 261797245138474 T^{3} + 22563490300366186081 T^{4} )^{2}$$
$43$ $$1 + 293538 T + 43082278722 T^{2} + 5003931048484746 T^{3} +$$$$46\!\cdots\!54$$$$T^{4} +$$$$31\!\cdots\!54$$$$T^{5} +$$$$17\!\cdots\!22$$$$T^{6} +$$$$74\!\cdots\!62$$$$T^{7} +$$$$15\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 195438 T + 19098005922 T^{2} - 2028906606479286 T^{3} +$$$$21\!\cdots\!94$$$$T^{4} -$$$$21\!\cdots\!94$$$$T^{5} +$$$$22\!\cdots\!02$$$$T^{6} -$$$$24\!\cdots\!82$$$$T^{7} +$$$$13\!\cdots\!81$$$$T^{8}$$
$53$ $$1 + 121988 T + 7440536072 T^{2} - 1624086878501364 T^{3} -$$$$90\!\cdots\!06$$$$T^{4} -$$$$35\!\cdots\!56$$$$T^{5} +$$$$36\!\cdots\!52$$$$T^{6} +$$$$13\!\cdots\!32$$$$T^{7} +$$$$24\!\cdots\!81$$$$T^{8}$$
$59$ $$1 - 112012206164 T^{2} +$$$$61\!\cdots\!86$$$$T^{4} -$$$$19\!\cdots\!84$$$$T^{6} +$$$$31\!\cdots\!61$$$$T^{8}$$
$61$ $$( 1 + 121686 T + 39182323346 T^{2} + 6269308274492646 T^{3} +$$$$26\!\cdots\!21$$$$T^{4} )^{2}$$
$67$ $$1 + 773602 T + 299230027202 T^{2} + 89051195209167114 T^{3} +$$$$25\!\cdots\!54$$$$T^{4} +$$$$80\!\cdots\!66$$$$T^{5} +$$$$24\!\cdots\!22$$$$T^{6} +$$$$57\!\cdots\!18$$$$T^{7} +$$$$66\!\cdots\!21$$$$T^{8}$$
$71$ $$( 1 + 523066 T + 324503556906 T^{2} + 67004903109421786 T^{3} +$$$$16\!\cdots\!41$$$$T^{4} )^{2}$$
$73$ $$1 - 922372 T + 425385053192 T^{2} - 220564608558389964 T^{3} +$$$$10\!\cdots\!34$$$$T^{4} -$$$$33\!\cdots\!96$$$$T^{5} +$$$$97\!\cdots\!32$$$$T^{6} -$$$$31\!\cdots\!68$$$$T^{7} +$$$$52\!\cdots\!41$$$$T^{8}$$
$79$ $$1 - 860907262084 T^{2} +$$$$30\!\cdots\!46$$$$T^{4} -$$$$50\!\cdots\!44$$$$T^{6} +$$$$34\!\cdots\!81$$$$T^{8}$$
$83$ $$1 + 1238058 T + 766393805682 T^{2} + 555939204099345666 T^{3} +$$$$38\!\cdots\!94$$$$T^{4} +$$$$18\!\cdots\!54$$$$T^{5} +$$$$81\!\cdots\!02$$$$T^{6} +$$$$43\!\cdots\!22$$$$T^{7} +$$$$11\!\cdots\!21$$$$T^{8}$$
$89$ $$1 - 1948062879044 T^{2} +$$$$14\!\cdots\!26$$$$T^{4} -$$$$48\!\cdots\!24$$$$T^{6} +$$$$61\!\cdots\!41$$$$T^{8}$$
$97$ $$1 + 1937532 T + 1877015125512 T^{2} + 2272282709664414324 T^{3} +$$$$26\!\cdots\!14$$$$T^{4} +$$$$18\!\cdots\!96$$$$T^{5} +$$$$13\!\cdots\!92$$$$T^{6} +$$$$11\!\cdots\!48$$$$T^{7} +$$$$48\!\cdots\!81$$$$T^{8}$$