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 $4$

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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:

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 \(S_{2}^{\mathrm{new}}(900, [\chi])\):

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