# Properties

 Label 900.2.bj.f Level $900$ Weight $2$ Character orbit 900.bj Analytic conductor $7.187$ Analytic rank $0$ Dimension $240$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 900.bj (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.18653618192$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$30$$ over $$\Q(\zeta_{20})$$ Twist minimal: no (minimal twist has level 300) Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$240q - 12q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$240q - 12q^{8} + 8q^{10} + 4q^{13} - 20q^{17} + 20q^{20} - 12q^{22} + 20q^{25} + 4q^{28} + 20q^{32} - 4q^{37} + 76q^{38} - 92q^{40} + 140q^{44} + 164q^{50} - 172q^{52} + 4q^{53} - 120q^{58} + 44q^{62} - 60q^{64} + 20q^{65} - 16q^{68} - 44q^{70} - 44q^{73} + 48q^{77} + 4q^{80} + 24q^{82} - 64q^{85} + 60q^{88} + 260q^{89} - 144q^{92} + 40q^{94} - 180q^{97} - 256q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1 −1.41129 + 0.0909451i 0 1.98346 0.256699i 1.83309 1.28054i 0 1.46352 1.46352i −2.77588 + 0.542662i 0 −2.47056 + 1.97391i
127.2 −1.36363 + 0.374857i 0 1.71896 1.02233i −2.16584 + 0.556007i 0 −3.13412 + 3.13412i −1.96080 + 2.03845i 0 2.74498 1.57007i
127.3 −1.36070 + 0.385348i 0 1.70301 1.04869i 0.0577937 + 2.23532i 0 −0.0211854 + 0.0211854i −1.91318 + 2.08320i 0 −0.940016 3.01933i
127.4 −1.33300 + 0.472333i 0 1.55380 1.25924i −1.01218 1.99386i 0 0.189838 0.189838i −1.47645 + 2.41249i 0 2.29101 + 2.17974i
127.5 −1.31411 0.522605i 0 1.45377 + 1.37352i 1.83309 1.28054i 0 −1.46352 + 1.46352i −1.19260 2.56470i 0 −3.07810 + 0.724779i
127.6 −1.18105 0.777895i 0 0.789760 + 1.83747i −2.16584 + 0.556007i 0 3.13412 3.13412i 0.496609 2.78449i 0 2.99048 + 1.02812i
127.7 −1.17502 0.786967i 0 0.761365 + 1.84941i 0.0577937 + 2.23532i 0 0.0211854 0.0211854i 0.560803 2.77227i 0 1.69122 2.67204i
127.8 −1.12180 0.861136i 0 0.516889 + 1.93205i −1.01218 1.99386i 0 −0.189838 + 0.189838i 1.08391 2.61250i 0 −0.581516 + 3.10835i
127.9 −0.872257 + 1.11318i 0 −0.478336 1.94196i 2.16662 0.552943i 0 2.97520 2.97520i 2.57898 + 1.16141i 0 −1.27433 + 2.89415i
127.10 −0.729889 + 1.21131i 0 −0.934525 1.76824i −1.31656 1.80739i 0 1.89964 1.89964i 2.82398 + 0.158622i 0 3.15025 0.275559i
127.11 −0.510631 + 1.31881i 0 −1.47851 1.34685i 1.27726 + 1.83538i 0 −3.58459 + 3.58459i 2.53121 1.26213i 0 −3.07272 + 0.747264i
127.12 −0.485574 1.32824i 0 −1.52844 + 1.28992i 2.16662 0.552943i 0 −2.97520 + 2.97520i 2.45549 + 1.40378i 0 −1.78650 2.60930i
127.13 −0.319851 1.37757i 0 −1.79539 + 0.881234i −1.31656 1.80739i 0 −1.89964 + 1.89964i 1.78822 + 2.19141i 0 −2.06871 + 2.39175i
127.14 −0.262182 + 1.38970i 0 −1.86252 0.728708i −0.583536 + 2.15858i 0 −0.889008 + 0.889008i 1.50100 2.39729i 0 −2.84679 1.37688i
127.15 −0.0781043 1.41206i 0 −1.98780 + 0.220575i 1.27726 + 1.83538i 0 3.58459 3.58459i 0.466720 + 2.78965i 0 2.49189 1.94691i
127.16 0.0475977 + 1.41341i 0 −1.99547 + 0.134550i −2.23568 + 0.0415918i 0 1.15642 1.15642i −0.285155 2.81402i 0 −0.165200 3.15796i
127.17 0.180090 1.40270i 0 −1.93514 0.505225i −0.583536 + 2.15858i 0 0.889008 0.889008i −1.05718 + 2.62343i 0 2.92276 + 1.20727i
127.18 0.482036 1.32953i 0 −1.53528 1.28176i −2.23568 + 0.0415918i 0 −1.15642 + 1.15642i −2.44420 + 1.42334i 0 −1.02238 + 2.99245i
127.19 0.510574 + 1.31883i 0 −1.47863 + 1.34672i −0.968294 2.01554i 0 −2.67499 + 2.67499i −2.53105 1.26246i 0 2.16377 2.30610i
127.20 0.753043 + 1.19705i 0 −0.865853 + 1.80286i −1.25978 + 1.84742i 0 1.55959 1.55959i −2.81013 + 0.321160i 0 −3.16012 0.116836i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 883.30 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.f odd 20 1 inner
100.l even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.bj.f 240
3.b odd 2 1 300.2.w.a 240
4.b odd 2 1 inner 900.2.bj.f 240
12.b even 2 1 300.2.w.a 240
25.f odd 20 1 inner 900.2.bj.f 240
75.l even 20 1 300.2.w.a 240
100.l even 20 1 inner 900.2.bj.f 240
300.u odd 20 1 300.2.w.a 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.w.a 240 3.b odd 2 1
300.2.w.a 240 12.b even 2 1
300.2.w.a 240 75.l even 20 1
300.2.w.a 240 300.u odd 20 1
900.2.bj.f 240 1.a even 1 1 trivial
900.2.bj.f 240 4.b odd 2 1 inner
900.2.bj.f 240 25.f odd 20 1 inner
900.2.bj.f 240 100.l even 20 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(900, [\chi])$$:

 $$86\!\cdots\!75$$$$T_{7}^{224} +$$$$10\!\cdots\!00$$$$T_{7}^{220} +$$$$93\!\cdots\!90$$$$T_{7}^{216} +$$$$70\!\cdots\!00$$$$T_{7}^{212} +$$$$43\!\cdots\!25$$$$T_{7}^{208} +$$$$22\!\cdots\!40$$$$T_{7}^{204} +$$$$98\!\cdots\!60$$$$T_{7}^{200} +$$$$36\!\cdots\!40$$$$T_{7}^{196} +$$$$11\!\cdots\!30$$$$T_{7}^{192} +$$$$32\!\cdots\!40$$$$T_{7}^{188} +$$$$77\!\cdots\!40$$$$T_{7}^{184} +$$$$16\!\cdots\!04$$$$T_{7}^{180} +$$$$28\!\cdots\!80$$$$T_{7}^{176} +$$$$45\!\cdots\!80$$$$T_{7}^{172} +$$$$62\!\cdots\!80$$$$T_{7}^{168} +$$$$74\!\cdots\!00$$$$T_{7}^{164} +$$$$77\!\cdots\!00$$$$T_{7}^{160} +$$$$70\!\cdots\!00$$$$T_{7}^{156} +$$$$56\!\cdots\!00$$$$T_{7}^{152} +$$$$38\!\cdots\!20$$$$T_{7}^{148} +$$$$23\!\cdots\!35$$$$T_{7}^{144} +$$$$11\!\cdots\!00$$$$T_{7}^{140} +$$$$53\!\cdots\!70$$$$T_{7}^{136} +$$$$20\!\cdots\!00$$$$T_{7}^{132} +$$$$69\!\cdots\!85$$$$T_{7}^{128} +$$$$19\!\cdots\!60$$$$T_{7}^{124} +$$$$48\!\cdots\!26$$$$T_{7}^{120} +$$$$10\!\cdots\!60$$$$T_{7}^{116} +$$$$18\!\cdots\!20$$$$T_{7}^{112} +$$$$28\!\cdots\!60$$$$T_{7}^{108} +$$$$36\!\cdots\!50$$$$T_{7}^{104} +$$$$40\!\cdots\!00$$$$T_{7}^{100} +$$$$37\!\cdots\!05$$$$T_{7}^{96} +$$$$28\!\cdots\!00$$$$T_{7}^{92} +$$$$18\!\cdots\!90$$$$T_{7}^{88} +$$$$10\!\cdots\!00$$$$T_{7}^{84} +$$$$44\!\cdots\!45$$$$T_{7}^{80} +$$$$16\!\cdots\!20$$$$T_{7}^{76} +$$$$48\!\cdots\!60$$$$T_{7}^{72} +$$$$11\!\cdots\!00$$$$T_{7}^{68} +$$$$21\!\cdots\!00$$$$T_{7}^{64} +$$$$31\!\cdots\!44$$$$T_{7}^{60} +$$$$35\!\cdots\!80$$$$T_{7}^{56} +$$$$28\!\cdots\!20$$$$T_{7}^{52} +$$$$15\!\cdots\!80$$$$T_{7}^{48} +$$$$59\!\cdots\!00$$$$T_{7}^{44} +$$$$14\!\cdots\!00$$$$T_{7}^{40} +$$$$21\!\cdots\!20$$$$T_{7}^{36} +$$$$19\!\cdots\!00$$$$T_{7}^{32} +$$$$10\!\cdots\!20$$$$T_{7}^{28} +$$$$33\!\cdots\!80$$$$T_{7}^{24} +$$$$60\!\cdots\!80$$$$T_{7}^{20} +$$$$57\!\cdots\!40$$$$T_{7}^{16} +$$$$25\!\cdots\!40$$$$T_{7}^{12} +$$$$42\!\cdots\!80$$$$T_{7}^{8} +$$$$65\!\cdots\!80$$$$T_{7}^{4} +$$$$52\!\cdots\!96$$">$$T_{7}^{240} + \cdots$$ $$10\!\cdots\!25$$$$T_{13}^{104} -$$$$52\!\cdots\!50$$$$T_{13}^{103} +$$$$15\!\cdots\!10$$$$T_{13}^{102} +$$$$35\!\cdots\!44$$$$T_{13}^{101} -$$$$10\!\cdots\!56$$$$T_{13}^{100} -$$$$79\!\cdots\!98$$$$T_{13}^{99} -$$$$41\!\cdots\!65$$$$T_{13}^{98} -$$$$28\!\cdots\!52$$$$T_{13}^{97} +$$$$82\!\cdots\!19$$$$T_{13}^{96} +$$$$19\!\cdots\!08$$$$T_{13}^{95} +$$$$40\!\cdots\!76$$$$T_{13}^{94} +$$$$17\!\cdots\!40$$$$T_{13}^{93} -$$$$54\!\cdots\!04$$$$T_{13}^{92} -$$$$10\!\cdots\!48$$$$T_{13}^{91} -$$$$29\!\cdots\!08$$$$T_{13}^{90} -$$$$60\!\cdots\!36$$$$T_{13}^{89} +$$$$32\!\cdots\!79$$$$T_{13}^{88} +$$$$14\!\cdots\!46$$$$T_{13}^{87} +$$$$21\!\cdots\!26$$$$T_{13}^{86} -$$$$60\!\cdots\!84$$$$T_{13}^{85} -$$$$14\!\cdots\!44$$$$T_{13}^{84} -$$$$11\!\cdots\!62$$$$T_{13}^{83} -$$$$84\!\cdots\!57$$$$T_{13}^{82} +$$$$31\!\cdots\!24$$$$T_{13}^{81} +$$$$51\!\cdots\!77$$$$T_{13}^{80} +$$$$73\!\cdots\!84$$$$T_{13}^{79} +$$$$15\!\cdots\!84$$$$T_{13}^{78} -$$$$18\!\cdots\!16$$$$T_{13}^{77} -$$$$15\!\cdots\!36$$$$T_{13}^{76} -$$$$26\!\cdots\!80$$$$T_{13}^{75} +$$$$10\!\cdots\!82$$$$T_{13}^{74} +$$$$61\!\cdots\!24$$$$T_{13}^{73} +$$$$40\!\cdots\!73$$$$T_{13}^{72} +$$$$41\!\cdots\!78$$$$T_{13}^{71} -$$$$74\!\cdots\!10$$$$T_{13}^{70} -$$$$17\!\cdots\!52$$$$T_{13}^{69} -$$$$67\!\cdots\!72$$$$T_{13}^{68} -$$$$61\!\cdots\!94$$$$T_{13}^{67} +$$$$40\!\cdots\!71$$$$T_{13}^{66} +$$$$30\!\cdots\!84$$$$T_{13}^{65} +$$$$94\!\cdots\!49$$$$T_{13}^{64} +$$$$58\!\cdots\!84$$$$T_{13}^{63} -$$$$80\!\cdots\!52$$$$T_{13}^{62} -$$$$42\!\cdots\!28$$$$T_{13}^{61} -$$$$93\!\cdots\!32$$$$T_{13}^{60} +$$$$78\!\cdots\!52$$$$T_{13}^{59} +$$$$10\!\cdots\!98$$$$T_{13}^{58} +$$$$41\!\cdots\!28$$$$T_{13}^{57} +$$$$65\!\cdots\!56$$$$T_{13}^{56} -$$$$13\!\cdots\!08$$$$T_{13}^{55} -$$$$98\!\cdots\!28$$$$T_{13}^{54} -$$$$28\!\cdots\!84$$$$T_{13}^{53} -$$$$28\!\cdots\!84$$$$T_{13}^{52} +$$$$13\!\cdots\!32$$$$T_{13}^{51} +$$$$64\!\cdots\!66$$$$T_{13}^{50} +$$$$14\!\cdots\!00$$$$T_{13}^{49} +$$$$61\!\cdots\!48$$$$T_{13}^{48} -$$$$87\!\cdots\!68$$$$T_{13}^{47} -$$$$31\!\cdots\!44$$$$T_{13}^{46} -$$$$60\!\cdots\!96$$$$T_{13}^{45} -$$$$15\!\cdots\!20$$$$T_{13}^{44} +$$$$33\!\cdots\!80$$$$T_{13}^{43} +$$$$11\!\cdots\!30$$$$T_{13}^{42} +$$$$23\!\cdots\!56$$$$T_{13}^{41} +$$$$23\!\cdots\!26$$$$T_{13}^{40} -$$$$36\!\cdots\!92$$$$T_{13}^{39} -$$$$14\!\cdots\!80$$$$T_{13}^{38} -$$$$17\!\cdots\!28$$$$T_{13}^{37} +$$$$43\!\cdots\!76$$$$T_{13}^{36} +$$$$36\!\cdots\!32$$$$T_{13}^{35} +$$$$11\!\cdots\!04$$$$T_{13}^{34} +$$$$24\!\cdots\!20$$$$T_{13}^{33} +$$$$35\!\cdots\!44$$$$T_{13}^{32} +$$$$29\!\cdots\!68$$$$T_{13}^{31} +$$$$59\!\cdots\!08$$$$T_{13}^{30} -$$$$53\!\cdots\!24$$$$T_{13}^{29} -$$$$19\!\cdots\!04$$$$T_{13}^{28} -$$$$43\!\cdots\!56$$$$T_{13}^{27} -$$$$11\!\cdots\!16$$$$T_{13}^{26} +$$$$44\!\cdots\!64$$$$T_{13}^{25} +$$$$15\!\cdots\!09$$$$T_{13}^{24} +$$$$63\!\cdots\!02$$$$T_{13}^{23} -$$$$32\!\cdots\!58$$$$T_{13}^{22} -$$$$13\!\cdots\!24$$$$T_{13}^{21} +$$$$30\!\cdots\!48$$$$T_{13}^{20} -$$$$81\!\cdots\!34$$$$T_{13}^{19} -$$$$12\!\cdots\!99$$$$T_{13}^{18} +$$$$14\!\cdots\!56$$$$T_{13}^{17} +$$$$23\!\cdots\!21$$$$T_{13}^{16} -$$$$24\!\cdots\!40$$$$T_{13}^{15} -$$$$68\!\cdots\!12$$$$T_{13}^{14} -$$$$22\!\cdots\!84$$$$T_{13}^{13} -$$$$18\!\cdots\!28$$$$T_{13}^{12} +$$$$14\!\cdots\!32$$$$T_{13}^{11} -$$$$27\!\cdots\!60$$$$T_{13}^{10} +$$$$10\!\cdots\!32$$$$T_{13}^{9} +$$$$13\!\cdots\!12$$$$T_{13}^{8} -$$$$18\!\cdots\!56$$$$T_{13}^{7} +$$$$13\!\cdots\!24$$$$T_{13}^{6} -$$$$59\!\cdots\!44$$$$T_{13}^{5} +$$$$18\!\cdots\!56$$$$T_{13}^{4} -$$$$43\!\cdots\!44$$$$T_{13}^{3} +$$$$81\!\cdots\!92$$$$T_{13}^{2} -$$$$67\!\cdots\!92$$$$T_{13} +$$$$32\!\cdots\!36$$">$$T_{13}^{120} - \cdots$$ $$37\!\cdots\!45$$$$T_{17}^{106} -$$$$18\!\cdots\!20$$$$T_{17}^{105} -$$$$11\!\cdots\!60$$$$T_{17}^{104} +$$$$19\!\cdots\!90$$$$T_{17}^{103} +$$$$38\!\cdots\!40$$$$T_{17}^{102} +$$$$22\!\cdots\!20$$$$T_{17}^{101} +$$$$13\!\cdots\!64$$$$T_{17}^{100} +$$$$12\!\cdots\!70$$$$T_{17}^{99} -$$$$27\!\cdots\!75$$$$T_{17}^{98} -$$$$17\!\cdots\!20$$$$T_{17}^{97} +$$$$72\!\cdots\!85$$$$T_{17}^{96} -$$$$10\!\cdots\!40$$$$T_{17}^{95} +$$$$41\!\cdots\!30$$$$T_{17}^{94} +$$$$27\!\cdots\!20$$$$T_{17}^{93} -$$$$72\!\cdots\!60$$$$T_{17}^{92} +$$$$61\!\cdots\!60$$$$T_{17}^{91} -$$$$15\!\cdots\!00$$$$T_{17}^{90} -$$$$22\!\cdots\!60$$$$T_{17}^{89} +$$$$15\!\cdots\!20$$$$T_{17}^{88} -$$$$64\!\cdots\!40$$$$T_{17}^{87} +$$$$13\!\cdots\!70$$$$T_{17}^{86} +$$$$25\!\cdots\!40$$$$T_{17}^{85} -$$$$17\!\cdots\!00$$$$T_{17}^{84} +$$$$74\!\cdots\!20$$$$T_{17}^{83} -$$$$66\!\cdots\!40$$$$T_{17}^{82} -$$$$22\!\cdots\!40$$$$T_{17}^{81} +$$$$16\!\cdots\!51$$$$T_{17}^{80} -$$$$64\!\cdots\!50$$$$T_{17}^{79} +$$$$36\!\cdots\!00$$$$T_{17}^{78} +$$$$17\!\cdots\!00$$$$T_{17}^{77} -$$$$12\!\cdots\!80$$$$T_{17}^{76} +$$$$45\!\cdots\!10$$$$T_{17}^{75} +$$$$22\!\cdots\!75$$$$T_{17}^{74} -$$$$12\!\cdots\!80$$$$T_{17}^{73} +$$$$84\!\cdots\!00$$$$T_{17}^{72} -$$$$30\!\cdots\!90$$$$T_{17}^{71} +$$$$19\!\cdots\!00$$$$T_{17}^{70} +$$$$52\!\cdots\!40$$$$T_{17}^{69} -$$$$37\!\cdots\!60$$$$T_{17}^{68} +$$$$13\!\cdots\!10$$$$T_{17}^{67} -$$$$23\!\cdots\!75$$$$T_{17}^{66} -$$$$90\!\cdots\!60$$$$T_{17}^{65} +$$$$88\!\cdots\!80$$$$T_{17}^{64} -$$$$37\!\cdots\!30$$$$T_{17}^{63} +$$$$10\!\cdots\!00$$$$T_{17}^{62} -$$$$10\!\cdots\!40$$$$T_{17}^{61} -$$$$44\!\cdots\!36$$$$T_{17}^{60} +$$$$27\!\cdots\!30$$$$T_{17}^{59} -$$$$76\!\cdots\!65$$$$T_{17}^{58} +$$$$57\!\cdots\!20$$$$T_{17}^{57} +$$$$50\!\cdots\!55$$$$T_{17}^{56} -$$$$27\!\cdots\!80$$$$T_{17}^{55} +$$$$71\!\cdots\!90$$$$T_{17}^{54} -$$$$21\!\cdots\!60$$$$T_{17}^{53} -$$$$56\!\cdots\!80$$$$T_{17}^{52} +$$$$21\!\cdots\!20$$$$T_{17}^{51} -$$$$33\!\cdots\!00$$$$T_{17}^{50} -$$$$63\!\cdots\!20$$$$T_{17}^{49} +$$$$48\!\cdots\!20$$$$T_{17}^{48} -$$$$10\!\cdots\!80$$$$T_{17}^{47} -$$$$24\!\cdots\!90$$$$T_{17}^{46} +$$$$98\!\cdots\!80$$$$T_{17}^{45} -$$$$22\!\cdots\!80$$$$T_{17}^{44} -$$$$16\!\cdots\!60$$$$T_{17}^{43} +$$$$15\!\cdots\!80$$$$T_{17}^{42} -$$$$20\!\cdots\!80$$$$T_{17}^{41} -$$$$41\!\cdots\!99$$$$T_{17}^{40} +$$$$13\!\cdots\!90$$$$T_{17}^{39} +$$$$15\!\cdots\!20$$$$T_{17}^{38} -$$$$23\!\cdots\!40$$$$T_{17}^{37} +$$$$11\!\cdots\!40$$$$T_{17}^{36} -$$$$94\!\cdots\!50$$$$T_{17}^{35} -$$$$74\!\cdots\!75$$$$T_{17}^{34} +$$$$45\!\cdots\!00$$$$T_{17}^{33} -$$$$20\!\cdots\!00$$$$T_{17}^{32} -$$$$16\!\cdots\!50$$$$T_{17}^{31} +$$$$57\!\cdots\!00$$$$T_{17}^{30} -$$$$11\!\cdots\!00$$$$T_{17}^{29} -$$$$16\!\cdots\!00$$$$T_{17}^{28} +$$$$12\!\cdots\!50$$$$T_{17}^{27} -$$$$11\!\cdots\!25$$$$T_{17}^{26} +$$$$48\!\cdots\!00$$$$T_{17}^{25} +$$$$42\!\cdots\!00$$$$T_{17}^{24} +$$$$34\!\cdots\!50$$$$T_{17}^{23} +$$$$52\!\cdots\!00$$$$T_{17}^{22} +$$$$48\!\cdots\!00$$$$T_{17}^{21} +$$$$21\!\cdots\!50$$$$T_{17}^{20} +$$$$17\!\cdots\!50$$$$T_{17}^{19} -$$$$36\!\cdots\!75$$$$T_{17}^{18} -$$$$55\!\cdots\!00$$$$T_{17}^{17} -$$$$53\!\cdots\!75$$$$T_{17}^{16} -$$$$65\!\cdots\!00$$$$T_{17}^{15} +$$$$12\!\cdots\!00$$$$T_{17}^{14} -$$$$14\!\cdots\!00$$$$T_{17}^{13} +$$$$10\!\cdots\!00$$$$T_{17}^{12} +$$$$17\!\cdots\!00$$$$T_{17}^{11} +$$$$43\!\cdots\!00$$$$T_{17}^{10} +$$$$14\!\cdots\!00$$$$T_{17}^{9} +$$$$25\!\cdots\!00$$$$T_{17}^{8} +$$$$50\!\cdots\!00$$$$T_{17}^{7} -$$$$24\!\cdots\!00$$$$T_{17}^{6} -$$$$25\!\cdots\!00$$$$T_{17}^{5} +$$$$97\!\cdots\!00$$$$T_{17}^{4} +$$$$63\!\cdots\!00$$$$T_{17}^{3} +$$$$21\!\cdots\!00$$$$T_{17}^{2} -$$$$56\!\cdots\!00$$$$T_{17} +$$$$26\!\cdots\!00$$">$$T_{17}^{120} + \cdots$$