# Properties

 Label 735.3.h.a Level 735 Weight 3 Character orbit 735.h Analytic conductor 20.027 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 735.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.0272994305$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.523596960000.16 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} -\beta_{5} q^{3} + ( 1 - \beta_{1} + \beta_{3} ) q^{4} -\beta_{6} q^{5} + ( -\beta_{2} + \beta_{5} ) q^{6} + ( -3 + 2 \beta_{1} - \beta_{3} + \beta_{7} ) q^{8} -3 q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} -\beta_{5} q^{3} + ( 1 - \beta_{1} + \beta_{3} ) q^{4} -\beta_{6} q^{5} + ( -\beta_{2} + \beta_{5} ) q^{6} + ( -3 + 2 \beta_{1} - \beta_{3} + \beta_{7} ) q^{8} -3 q^{9} + ( \beta_{2} - \beta_{4} + \beta_{6} ) q^{10} + ( -6 - \beta_{1} + \beta_{3} - \beta_{7} ) q^{11} + ( \beta_{2} - 2 \beta_{5} + 3 \beta_{6} ) q^{12} + ( -2 \beta_{2} + 4 \beta_{4} + \beta_{6} ) q^{13} -\beta_{1} q^{15} + ( -3 - \beta_{1} + 5 \beta_{3} - 2 \beta_{7} ) q^{16} + ( -4 \beta_{2} + \beta_{4} - 2 \beta_{6} ) q^{17} + 3 \beta_{3} q^{18} + ( 2 \beta_{2} - 4 \beta_{4} + \beta_{5} + 8 \beta_{6} ) q^{19} + ( -\beta_{2} + \beta_{4} + 5 \beta_{5} - 2 \beta_{6} ) q^{20} + ( -3 + 4 \beta_{1} + 3 \beta_{3} + \beta_{7} ) q^{22} + ( -12 + 2 \beta_{1} - 3 \beta_{3} + 4 \beta_{7} ) q^{23} + ( -\beta_{2} + 3 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} ) q^{24} -5 q^{25} + ( -\beta_{2} + \beta_{4} - 10 \beta_{5} + 13 \beta_{6} ) q^{26} + 3 \beta_{5} q^{27} + ( -15 + 4 \beta_{1} - 5 \beta_{7} ) q^{29} + ( 2 + \beta_{1} - 3 \beta_{3} + \beta_{7} ) q^{30} + ( \beta_{2} + 4 \beta_{4} - 13 \beta_{5} + 7 \beta_{6} ) q^{31} + ( -11 + 2 \beta_{1} + \beta_{3} - 3 \beta_{7} ) q^{32} + ( \beta_{2} - 3 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} ) q^{33} + ( 2 \beta_{2} - 2 \beta_{4} - 20 \beta_{5} + 16 \beta_{6} ) q^{34} + ( -3 + 3 \beta_{1} - 3 \beta_{3} ) q^{36} + ( 22 - 7 \beta_{1} - 8 \beta_{3} - 4 \beta_{7} ) q^{37} + ( -7 \beta_{2} + 8 \beta_{4} + 9 \beta_{5} - 22 \beta_{6} ) q^{38} + ( -2 + \beta_{1} + 6 \beta_{3} - 4 \beta_{7} ) q^{39} + ( 3 \beta_{2} + 2 \beta_{4} - 10 \beta_{5} + 3 \beta_{6} ) q^{40} + ( -3 \beta_{2} + 3 \beta_{4} + 4 \beta_{5} + 9 \beta_{6} ) q^{41} + ( 39 - 3 \beta_{1} + 5 \beta_{3} - 5 \beta_{7} ) q^{43} + ( 1 + \beta_{1} + 7 \beta_{3} ) q^{44} + 3 \beta_{6} q^{45} + ( 11 - 13 \beta_{1} + 17 \beta_{3} - 2 \beta_{7} ) q^{46} + ( \beta_{2} + \beta_{4} + 5 \beta_{5} + 4 \beta_{6} ) q^{47} + ( 5 \beta_{2} - 6 \beta_{4} + 3 \beta_{6} ) q^{48} + 5 \beta_{3} q^{50} + ( -11 - 2 \beta_{1} + 12 \beta_{3} - \beta_{7} ) q^{51} + ( -15 \beta_{2} - 3 \beta_{4} + 5 \beta_{5} - 12 \beta_{6} ) q^{52} + ( 23 + 12 \beta_{1} - 3 \beta_{3} + 5 \beta_{7} ) q^{53} + ( 3 \beta_{2} - 3 \beta_{5} ) q^{54} + ( -3 \beta_{2} - 2 \beta_{4} + 5 \beta_{5} + 6 \beta_{6} ) q^{55} + ( 5 + 8 \beta_{1} - 6 \beta_{3} + 4 \beta_{7} ) q^{57} + ( -8 + 6 \beta_{1} + 32 \beta_{3} - 4 \beta_{7} ) q^{58} + ( 14 \beta_{2} + \beta_{4} - 12 \beta_{5} - 14 \beta_{6} ) q^{59} + ( 13 - 2 \beta_{1} + 3 \beta_{3} - \beta_{7} ) q^{60} + ( 4 \beta_{2} - 10 \beta_{4} + 36 \beta_{5} + 14 \beta_{6} ) q^{61} + ( -20 \beta_{2} + 7 \beta_{4} + 18 \beta_{5} - 2 \beta_{6} ) q^{62} + ( 3 + 9 \beta_{1} - \beta_{3} + 6 \beta_{7} ) q^{64} + ( 21 - 14 \beta_{3} - 2 \beta_{7} ) q^{65} + ( 3 \beta_{2} + 3 \beta_{4} - \beta_{5} - 12 \beta_{6} ) q^{66} + ( -49 + 9 \beta_{1} + 5 \beta_{3} - \beta_{7} ) q^{67} + ( -20 \beta_{2} + 12 \beta_{4} + 30 \beta_{5} - 18 \beta_{6} ) q^{68} + ( -3 \beta_{2} + 12 \beta_{4} + 11 \beta_{5} - 6 \beta_{6} ) q^{69} + ( 9 + 4 \beta_{1} + 28 \beta_{3} + 5 \beta_{7} ) q^{71} + ( 9 - 6 \beta_{1} + 3 \beta_{3} - 3 \beta_{7} ) q^{72} + ( 11 \beta_{2} - 7 \beta_{4} + 27 \beta_{5} + 23 \beta_{6} ) q^{73} + ( 54 + 7 \beta_{1} - 31 \beta_{3} + 7 \beta_{7} ) q^{74} + 5 \beta_{5} q^{75} + ( 23 \beta_{2} - 6 \beta_{4} - 48 \beta_{5} + 27 \beta_{6} ) q^{76} + ( -32 + 13 \beta_{1} + 3 \beta_{3} - \beta_{7} ) q^{78} + ( 52 + 9 \beta_{1} + 5 \beta_{3} + 6 \beta_{7} ) q^{79} + ( -9 \beta_{2} - \beta_{4} + 5 \beta_{5} ) q^{80} + 9 q^{81} + ( -5 \beta_{2} + 9 \beta_{4} - 19 \beta_{5} + 6 \beta_{6} ) q^{82} + ( 12 \beta_{2} + 5 \beta_{4} - 8 \beta_{5} + 32 \beta_{6} ) q^{83} + ( 1 - 14 \beta_{3} + 3 \beta_{7} ) q^{85} + ( -19 + 18 \beta_{1} - 48 \beta_{3} + 3 \beta_{7} ) q^{86} + ( -15 \beta_{4} + 20 \beta_{5} - 12 \beta_{6} ) q^{87} + ( -25 - 10 \beta_{1} - 17 \beta_{3} - 5 \beta_{7} ) q^{88} + ( -10 \beta_{2} + 11 \beta_{4} - 24 \beta_{5} - 22 \beta_{6} ) q^{89} + ( -3 \beta_{2} + 3 \beta_{4} - 3 \beta_{6} ) q^{90} + ( -11 + 26 \beta_{1} - 53 \beta_{3} - 3 \beta_{7} ) q^{92} + ( -32 + 7 \beta_{1} - 3 \beta_{3} - 4 \beta_{7} ) q^{93} + ( \beta_{2} + 4 \beta_{4} - 5 \beta_{6} ) q^{94} + ( 24 + \beta_{1} + 14 \beta_{3} + 2 \beta_{7} ) q^{95} + ( \beta_{2} - 9 \beta_{4} + 13 \beta_{5} - 6 \beta_{6} ) q^{96} + ( -14 \beta_{2} - 10 \beta_{4} + 30 \beta_{5} + 26 \beta_{6} ) q^{97} + ( 18 + 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{2} + 12q^{4} - 32q^{8} - 24q^{9} + O(q^{10})$$ $$8q - 4q^{2} + 12q^{4} - 32q^{8} - 24q^{9} - 40q^{11} + 4q^{16} + 12q^{18} - 16q^{22} - 124q^{23} - 40q^{25} - 100q^{29} - 72q^{32} - 36q^{36} + 160q^{37} + 24q^{39} + 352q^{43} + 36q^{44} + 164q^{46} + 20q^{50} - 36q^{51} + 152q^{53} + 80q^{58} + 120q^{60} - 4q^{64} + 120q^{65} - 368q^{67} + 164q^{71} + 96q^{72} + 280q^{74} - 240q^{78} + 412q^{79} + 72q^{81} - 60q^{85} - 356q^{86} - 248q^{88} - 288q^{92} - 252q^{93} + 240q^{95} + 120q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 13 x^{6} - 2 x^{5} + 91 x^{4} - 50 x^{3} + 190 x^{2} + 100 x + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$9 \nu^{7} - 38 \nu^{6} + 87 \nu^{5} + 132 \nu^{4} - 161 \nu^{3} + 480 \nu^{2} + 300 \nu + 12250$$$$)/3090$$ $$\beta_{2}$$ $$=$$ $$($$$$87 \nu^{7} - 24 \nu^{6} + 841 \nu^{5} + 1276 \nu^{4} + 10117 \nu^{3} + 4640 \nu^{2} + 46160 \nu + 13700$$$$)/21630$$ $$\beta_{3}$$ $$=$$ $$($$$$87 \nu^{7} - 24 \nu^{6} + 841 \nu^{5} + 1276 \nu^{4} + 10117 \nu^{3} + 4640 \nu^{2} + 2900 \nu + 35330$$$$)/21630$$ $$\beta_{4}$$ $$=$$ $$($$$$283 \nu^{7} - 451 \nu^{6} + 5139 \nu^{5} - 656 \nu^{4} + 40658 \nu^{3} + 12690 \nu^{2} + 158440 \nu + 44150$$$$)/32445$$ $$\beta_{5}$$ $$=$$ $$($$$$-137 \nu^{7} + 361 \nu^{6} - 1805 \nu^{5} + 1115 \nu^{4} - 11191 \nu^{3} + 16967 \nu^{2} - 21390 \nu + 15$$$$)/10815$$ $$\beta_{6}$$ $$=$$ $$($$$$-11 \nu^{7} + 32 \nu^{6} - 153 \nu^{5} + 142 \nu^{4} - 901 \nu^{3} + 1350 \nu^{2} - 1580 \nu + 50$$$$)/630$$ $$\beta_{7}$$ $$=$$ $$($$$$423 \nu^{7} - 241 \nu^{6} + 4089 \nu^{5} + 6204 \nu^{4} + 34148 \nu^{3} + 22560 \nu^{2} + 14100 \nu + 61265$$$$)/10815$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{6} + 5 \beta_{5} + \beta_{3} + \beta_{2} + \beta_{1} - 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} + 9 \beta_{3} + \beta_{1} - 13$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{7} + 33 \beta_{6} - 45 \beta_{5} + 6 \beta_{4} + 17 \beta_{3} - 17 \beta_{2} + 11 \beta_{1} - 60$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$13 \beta_{7} + 63 \beta_{6} - 85 \beta_{5} + 39 \beta_{4} - 95 \beta_{3} - 95 \beta_{2} - 21 \beta_{1} + 167$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$34 \beta_{7} - 243 \beta_{3} - 121 \beta_{1} + 684$$ $$\nu^{7}$$ $$=$$ $$($$$$155 \beta_{7} - 933 \beta_{6} + 1215 \beta_{5} - 465 \beta_{4} - 1081 \beta_{3} + 1081 \beta_{2} - 311 \beta_{1} + 2141$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$$.

 $$n$$ $$346$$ $$442$$ $$491$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
391.1
 −1.26021 − 2.18275i −1.26021 + 2.18275i −0.336732 − 0.583237i −0.336732 + 0.583237i 0.836732 + 1.44926i 0.836732 − 1.44926i 1.76021 + 3.04878i 1.76021 − 3.04878i
−3.52043 1.73205i 8.39341 2.23607i 6.09756i 0 −15.4667 −3.00000 7.87192i
391.2 −3.52043 1.73205i 8.39341 2.23607i 6.09756i 0 −15.4667 −3.00000 7.87192i
391.3 −1.67346 1.73205i −1.19952 2.23607i 2.89852i 0 8.70121 −3.00000 3.74198i
391.4 −1.67346 1.73205i −1.19952 2.23607i 2.89852i 0 8.70121 −3.00000 3.74198i
391.5 0.673464 1.73205i −3.54645 2.23607i 1.16647i 0 −5.08226 −3.00000 1.50591i
391.6 0.673464 1.73205i −3.54645 2.23607i 1.16647i 0 −5.08226 −3.00000 1.50591i
391.7 2.52043 1.73205i 2.35256 2.23607i 4.36551i 0 −4.15226 −3.00000 5.63585i
391.8 2.52043 1.73205i 2.35256 2.23607i 4.36551i 0 −4.15226 −3.00000 5.63585i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 391.8 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.3.h.a 8
7.b odd 2 1 inner 735.3.h.a 8
7.c even 3 1 105.3.n.a 8
7.d odd 6 1 105.3.n.a 8
21.g even 6 1 315.3.w.a 8
21.h odd 6 1 315.3.w.a 8
35.i odd 6 1 525.3.o.l 8
35.j even 6 1 525.3.o.l 8
35.k even 12 2 525.3.s.h 16
35.l odd 12 2 525.3.s.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.n.a 8 7.c even 3 1
105.3.n.a 8 7.d odd 6 1
315.3.w.a 8 21.g even 6 1
315.3.w.a 8 21.h odd 6 1
525.3.o.l 8 35.i odd 6 1
525.3.o.l 8 35.j even 6 1
525.3.s.h 16 35.k even 12 2
525.3.s.h 16 35.l odd 12 2
735.3.h.a 8 1.a even 1 1 trivial
735.3.h.a 8 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 2 T_{2}^{3} - 9 T_{2}^{2} - 10 T_{2} + 10$$ acting on $$S_{3}^{\mathrm{new}}(735, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T + 7 T^{2} + 14 T^{3} + 34 T^{4} + 56 T^{5} + 112 T^{6} + 128 T^{7} + 256 T^{8} )^{2}$$
$3$ $$( 1 + 3 T^{2} )^{4}$$
$5$ $$( 1 + 5 T^{2} )^{4}$$
$7$ 1
$11$ $$( 1 + 20 T + 547 T^{2} + 6950 T^{3} + 103078 T^{4} + 840950 T^{5} + 8008627 T^{6} + 35431220 T^{7} + 214358881 T^{8} )^{2}$$
$13$ $$1 - 188 T^{2} + 39826 T^{4} - 8798048 T^{6} + 2113175419 T^{8} - 251281048928 T^{10} + 32487291694546 T^{12} - 4380040003026428 T^{14} + 665416609183179841 T^{16}$$
$17$ $$1 - 1208 T^{2} + 774016 T^{4} - 348305048 T^{6} + 116530286494 T^{8} - 29090785914008 T^{10} + 5399347871453056 T^{12} - 703807662573551288 T^{14} + 48661191875666868481 T^{16}$$
$19$ $$1 - 1196 T^{2} + 705850 T^{4} - 361350224 T^{6} + 155496381379 T^{8} - 47091522541904 T^{10} + 11987847972489850 T^{12} - 2647124643203128556 T^{14} +$$$$28\!\cdots\!81$$$$T^{16}$$
$23$ $$( 1 + 62 T + 2347 T^{2} + 75764 T^{3} + 2007934 T^{4} + 40079156 T^{5} + 656786827 T^{6} + 9178225118 T^{7} + 78310985281 T^{8} )^{2}$$
$29$ $$( 1 + 50 T + 1234 T^{2} - 15850 T^{3} - 1164374 T^{4} - 13329850 T^{5} + 872784754 T^{6} + 29741166050 T^{7} + 500246412961 T^{8} )^{2}$$
$31$ $$1 - 3890 T^{2} + 8819209 T^{4} - 13390045370 T^{6} + 15030304083796 T^{8} - 12365988090147770 T^{10} + 7521824313419004169 T^{12} -$$$$30\!\cdots\!90$$$$T^{14} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$( 1 - 80 T + 5206 T^{2} - 193760 T^{3} + 8139931 T^{4} - 265257440 T^{5} + 9756882166 T^{6} - 205258112720 T^{7} + 3512479453921 T^{8} )^{2}$$
$41$ $$1 - 10106 T^{2} + 48877645 T^{4} - 146585251874 T^{6} + 296639674915264 T^{8} - 414214887920726114 T^{10} +$$$$39\!\cdots\!45$$$$T^{12} -$$$$22\!\cdots\!86$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16}$$
$43$ $$( 1 - 176 T + 17017 T^{2} - 1139948 T^{3} + 56853640 T^{4} - 2107763852 T^{5} + 58177736617 T^{6} - 1112559896624 T^{7} + 11688200277601 T^{8} )^{2}$$
$47$ $$1 - 16718 T^{2} + 124152241 T^{4} - 535266039518 T^{6} + 1465949995987204 T^{8} - 2611927522981233758 T^{10} +$$$$29\!\cdots\!01$$$$T^{12} -$$$$19\!\cdots\!38$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$( 1 - 76 T + 8845 T^{2} - 557992 T^{3} + 33596848 T^{4} - 1567399528 T^{5} + 69791304445 T^{6} - 1684491445804 T^{7} + 62259690411361 T^{8} )^{2}$$
$59$ $$1 - 11480 T^{2} + 76423744 T^{4} - 376265621240 T^{6} + 1444840128344926 T^{8} - 4559346364454347640 T^{10} +$$$$11\!\cdots\!24$$$$T^{12} -$$$$20\!\cdots\!80$$$$T^{14} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$1 - 9512 T^{2} + 35469436 T^{4} - 58631449112 T^{6} + 50920349318854 T^{8} - 811801722004343192 T^{10} +$$$$67\!\cdots\!16$$$$T^{12} -$$$$25\!\cdots\!52$$$$T^{14} +$$$$36\!\cdots\!61$$$$T^{16}$$
$67$ $$( 1 + 184 T + 27625 T^{2} + 2611588 T^{3} + 208237768 T^{4} + 11723418532 T^{5} + 556674717625 T^{6} + 16644342319096 T^{7} + 406067677556641 T^{8} )^{2}$$
$71$ $$( 1 - 82 T + 12166 T^{2} - 846262 T^{3} + 94594474 T^{4} - 4266006742 T^{5} + 309158511046 T^{6} - 10504223281522 T^{7} + 645753531245761 T^{8} )^{2}$$
$73$ $$1 - 15422 T^{2} + 189396025 T^{4} - 1427786526086 T^{6} + 9122100606631828 T^{8} - 40546625864343014726 T^{10} +$$$$15\!\cdots\!25$$$$T^{12} -$$$$35\!\cdots\!62$$$$T^{14} +$$$$65\!\cdots\!61$$$$T^{16}$$
$79$ $$( 1 - 206 T + 36853 T^{2} - 4125578 T^{3} + 384207724 T^{4} - 25747732298 T^{5} + 1435427335093 T^{6} - 50076015837326 T^{7} + 1517108809906561 T^{8} )^{2}$$
$83$ $$1 - 20672 T^{2} + 223804480 T^{4} - 2182268545136 T^{6} + 17948924233578718 T^{8} -$$$$10\!\cdots\!56$$$$T^{10} +$$$$50\!\cdots\!80$$$$T^{12} -$$$$22\!\cdots\!92$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$1 - 39848 T^{2} + 820746880 T^{4} - 10893774296744 T^{6} + 101902892477590558 T^{8} -$$$$68\!\cdots\!04$$$$T^{10} +$$$$32\!\cdots\!80$$$$T^{12} -$$$$98\!\cdots\!08$$$$T^{14} +$$$$15\!\cdots\!61$$$$T^{16}$$
$97$ $$1 - 44576 T^{2} + 925514428 T^{4} - 12414040936928 T^{6} + 128325632901816454 T^{8} -$$$$10\!\cdots\!68$$$$T^{10} +$$$$72\!\cdots\!08$$$$T^{12} -$$$$30\!\cdots\!16$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16}$$