Properties

Label 160.6.n.d
Level 160
Weight 6
Character orbit 160.n
Analytic conductor 25.661
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 1375 x^{14} + 743087 x^{12} + 198706725 x^{10} + 26872635188 x^{8} + 1612811892960 x^{6} + 24971013117824 x^{4} + 124211190826240 x^{2} + 177426662425600\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{41}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{3} + \beta_{4} ) q^{3} + ( -3 + \beta_{4} - \beta_{8} ) q^{5} + ( -6 - \beta_{2} + 6 \beta_{4} - \beta_{8} + \beta_{10} ) q^{7} + ( -2 \beta_{2} - 2 \beta_{3} + 103 \beta_{4} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{3} + \beta_{4} ) q^{3} + ( -3 + \beta_{4} - \beta_{8} ) q^{5} + ( -6 - \beta_{2} + 6 \beta_{4} - \beta_{8} + \beta_{10} ) q^{7} + ( -2 \beta_{2} - 2 \beta_{3} + 103 \beta_{4} - \beta_{7} ) q^{9} + ( 2 \beta_{2} + 2 \beta_{3} + 26 \beta_{4} - \beta_{6} - \beta_{14} ) q^{11} + ( 35 + 2 \beta_{2} - 35 \beta_{4} - \beta_{6} + 2 \beta_{8} - \beta_{10} + \beta_{13} ) q^{13} + ( -44 + \beta_{1} - 93 \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{12} + \beta_{15} ) q^{15} + ( -116 + \beta_{1} - \beta_{3} - 115 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - 6 \beta_{8} + \beta_{11} - \beta_{15} ) q^{17} + ( 163 - 2 \beta_{1} + 8 \beta_{2} - 10 \beta_{3} - \beta_{4} + 3 \beta_{5} - 7 \beta_{6} - 6 \beta_{8} - \beta_{9} + 3 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{19} + ( -289 + 4 \beta_{1} - 19 \beta_{2} + 20 \beta_{3} + 2 \beta_{4} - \beta_{5} + 9 \beta_{6} - 7 \beta_{8} + 4 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{21} + ( -484 + 4 \beta_{1} - 4 \beta_{2} + 21 \beta_{3} - 488 \beta_{4} - 11 \beta_{5} - 8 \beta_{6} + 4 \beta_{7} - 3 \beta_{8} - \beta_{9} - 4 \beta_{12} + \beta_{14} ) q^{23} + ( 585 - 6 \beta_{1} + 39 \beta_{2} - 2 \beta_{3} - 253 \beta_{4} + 8 \beta_{5} - 6 \beta_{6} + \beta_{7} - \beta_{8} + 5 \beta_{10} + 4 \beta_{11} - \beta_{12} - \beta_{13} + 4 \beta_{14} ) q^{25} + ( -816 + 6 \beta_{1} - 98 \beta_{2} - 2 \beta_{3} + 818 \beta_{4} + 25 \beta_{6} - 6 \beta_{7} - 25 \beta_{8} + \beta_{9} + 8 \beta_{10} - 2 \beta_{11} - 4 \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{27} + ( 1 + 15 \beta_{2} + 23 \beta_{3} + 553 \beta_{4} - 16 \beta_{5} - 30 \beta_{6} - 6 \beta_{7} - 12 \beta_{8} + 16 \beta_{10} + 7 \beta_{11} - \beta_{12} - 4 \beta_{14} ) q^{29} + ( 11 + 30 \beta_{2} + 40 \beta_{3} + 869 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} - 10 \beta_{7} + 54 \beta_{8} - 10 \beta_{10} - \beta_{11} - 11 \beta_{12} + \beta_{13} - \beta_{15} ) q^{31} + ( 762 - 5 \beta_{1} - 9 \beta_{2} + 12 \beta_{3} - 747 \beta_{4} + 4 \beta_{6} + 5 \beta_{7} + 95 \beta_{8} - 4 \beta_{9} - 39 \beta_{10} + 12 \beta_{11} - 3 \beta_{12} - 5 \beta_{13} - 4 \beta_{14} ) q^{33} + ( 724 - 4 \beta_{1} + 83 \beta_{2} - 94 \beta_{3} - 1720 \beta_{4} + 7 \beta_{5} - 2 \beta_{6} + 14 \beta_{7} + 13 \beta_{8} - 25 \beta_{10} - 5 \beta_{11} - 7 \beta_{12} + \beta_{13} + \beta_{14} - 5 \beta_{15} ) q^{35} + ( -413 + 6 \beta_{1} - 5 \beta_{2} - 234 \beta_{3} - 404 \beta_{4} - 12 \beta_{5} - 19 \beta_{6} + 6 \beta_{7} - 67 \beta_{8} - 4 \beta_{9} + 14 \beta_{11} - 5 \beta_{12} + 4 \beta_{14} + 4 \beta_{15} ) q^{37} + ( 694 - 20 \beta_{1} + 63 \beta_{2} - 83 \beta_{3} + 2 \beta_{4} + 17 \beta_{5} - 9 \beta_{6} - 161 \beta_{8} + 17 \beta_{10} + 2 \beta_{11} - 22 \beta_{12} - 6 \beta_{13} - 6 \beta_{15} ) q^{39} + ( -1519 - 5 \beta_{1} - 201 \beta_{2} + 219 \beta_{3} + 19 \beta_{4} + 41 \beta_{5} + 35 \beta_{6} - 41 \beta_{8} - 16 \beta_{9} + 41 \beta_{10} + 19 \beta_{11} - \beta_{12} - 5 \beta_{13} - 5 \beta_{15} ) q^{41} + ( 1243 - 18 \beta_{1} - 10 \beta_{2} + 111 \beta_{3} + 1221 \beta_{4} - 32 \beta_{5} - 8 \beta_{6} - 18 \beta_{7} + 48 \beta_{8} - 12 \beta_{11} - 10 \beta_{12} - 2 \beta_{15} ) q^{43} + ( 47 + 291 \beta_{2} + 237 \beta_{3} - 293 \beta_{4} + 63 \beta_{5} - 112 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} - 4 \beta_{9} - 12 \beta_{10} + 23 \beta_{11} - 5 \beta_{12} + 4 \beta_{13} - 16 \beta_{14} + \beta_{15} ) q^{45} + ( 1376 - 14 \beta_{1} - 451 \beta_{2} + 2 \beta_{3} - 1334 \beta_{4} + 187 \beta_{6} + 14 \beta_{7} - 30 \beta_{8} - \beta_{9} + 11 \beta_{10} + 2 \beta_{11} - 40 \beta_{12} + 10 \beta_{13} - \beta_{14} ) q^{47} + ( 11 + 29 \beta_{2} + 69 \beta_{3} - 3416 \beta_{4} + 19 \beta_{5} - 175 \beta_{6} + 9 \beta_{7} + 63 \beta_{8} - 19 \beta_{10} + 29 \beta_{11} - 11 \beta_{12} - \beta_{13} + 16 \beta_{14} + \beta_{15} ) q^{49} + ( 13 + 330 \beta_{2} + 324 \beta_{3} - 1195 \beta_{4} + 5 \beta_{5} + 133 \beta_{6} + 22 \beta_{7} + 57 \beta_{8} - 5 \beta_{10} - 19 \beta_{11} - 13 \beta_{12} - 5 \beta_{13} + 14 \beta_{14} + 5 \beta_{15} ) q^{51} + ( 554 + 6 \beta_{1} - 689 \beta_{2} + 33 \beta_{3} - 514 \beta_{4} + 19 \beta_{6} - 6 \beta_{7} + 100 \beta_{8} + 16 \beta_{9} + 41 \beta_{10} + 33 \beta_{11} - 7 \beta_{12} - \beta_{13} + 16 \beta_{14} ) q^{53} + ( -402 + 15 \beta_{1} - 5 \beta_{2} - 760 \beta_{3} + 4995 \beta_{4} + 8 \beta_{5} - 5 \beta_{6} - 53 \beta_{7} - 29 \beta_{8} - 14 \beta_{9} - 17 \beta_{10} + 6 \beta_{11} - 61 \beta_{12} - 6 \beta_{13} - \beta_{14} + \beta_{15} ) q^{55} + ( 3673 - 16 \beta_{1} - 6 \beta_{2} - 679 \beta_{3} + 3702 \beta_{4} - 49 \beta_{5} - 31 \beta_{6} - 16 \beta_{7} - 196 \beta_{8} + 20 \beta_{9} + 35 \beta_{11} - 6 \beta_{12} - 20 \beta_{14} + 5 \beta_{15} ) q^{57} + ( -8813 + 78 \beta_{1} + 136 \beta_{2} - 170 \beta_{3} - 29 \beta_{4} - 7 \beta_{5} - 109 \beta_{6} - 28 \beta_{8} + 15 \beta_{9} - 7 \beta_{10} - 29 \beta_{11} - 5 \beta_{12} + 5 \beta_{13} + 5 \beta_{15} ) q^{59} + ( -1476 - 18 \beta_{1} - 234 \beta_{2} + 259 \beta_{3} + 37 \beta_{4} - 103 \beta_{5} + 251 \beta_{6} + 29 \beta_{8} - 103 \beta_{10} + 37 \beta_{11} - 12 \beta_{12} - \beta_{13} - \beta_{15} ) q^{61} + ( -7664 + 38 \beta_{1} - 82 \beta_{2} + 1001 \beta_{3} - 7726 \beta_{4} + 19 \beta_{5} - 454 \beta_{6} + 38 \beta_{7} - 187 \beta_{8} + 15 \beta_{9} + 20 \beta_{11} - 82 \beta_{12} - 15 \beta_{14} + 10 \beta_{15} ) q^{63} + ( -1904 + 55 \beta_{1} + 1371 \beta_{2} - 3 \beta_{3} + 4158 \beta_{4} - 175 \beta_{5} + 137 \beta_{6} - 10 \beta_{7} - 123 \beta_{8} + 20 \beta_{9} + 55 \beta_{10} + 37 \beta_{11} - 9 \beta_{12} + 5 \beta_{13} - 5 \beta_{15} ) q^{65} + ( -1199 - 4 \beta_{1} - 1421 \beta_{2} - 44 \beta_{3} + 1127 \beta_{4} - 110 \beta_{6} + 4 \beta_{7} - 168 \beta_{8} - 14 \beta_{9} - 10 \beta_{10} - 44 \beta_{11} + 28 \beta_{12} - 14 \beta_{14} ) q^{67} + ( 3 + 1189 \beta_{2} + 1216 \beta_{3} - 12524 \beta_{4} - 36 \beta_{5} - 112 \beta_{6} + 54 \beta_{7} - 24 \beta_{8} + 36 \beta_{10} + 24 \beta_{11} - 3 \beta_{12} + 4 \beta_{13} + 4 \beta_{14} - 4 \beta_{15} ) q^{69} + ( 87 + 774 \beta_{2} + 904 \beta_{3} + 2473 \beta_{4} - 58 \beta_{5} - 202 \beta_{6} + 30 \beta_{7} + 290 \beta_{8} + 58 \beta_{10} + 43 \beta_{11} - 87 \beta_{12} + \beta_{13} - \beta_{15} ) q^{71} + ( 695 + 40 \beta_{1} - 733 \beta_{2} - 2 \beta_{3} - 698 \beta_{4} - 40 \beta_{6} - 40 \beta_{7} + 25 \beta_{8} + 4 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + \beta_{12} + 41 \beta_{13} + 4 \beta_{14} ) q^{73} + ( 16920 - 46 \beta_{1} + 441 \beta_{2} - 1838 \beta_{3} + 3176 \beta_{4} - 35 \beta_{5} + 48 \beta_{6} + 4 \beta_{7} - 56 \beta_{8} - \beta_{9} + 157 \beta_{10} - 53 \beta_{11} + 67 \beta_{12} + 5 \beta_{13} - 15 \beta_{14} + 29 \beta_{15} ) q^{75} + ( 9428 - 56 \beta_{1} + 11 \beta_{2} - 2982 \beta_{3} + 9443 \beta_{4} + 235 \beta_{5} - 203 \beta_{6} - 56 \beta_{7} - 28 \beta_{8} - 16 \beta_{9} + 4 \beta_{11} + 11 \beta_{12} + 16 \beta_{14} - 35 \beta_{15} ) q^{77} + ( -9599 + 30 \beta_{1} + 699 \beta_{2} - 749 \beta_{3} + 41 \beta_{4} - 83 \beta_{5} + 247 \beta_{6} - 393 \beta_{8} - 83 \beta_{10} + 41 \beta_{11} - 91 \beta_{12} + 35 \beta_{13} + 35 \beta_{15} ) q^{79} + ( -16276 + 19 \beta_{1} - 1937 \beta_{2} + 1899 \beta_{3} - 17 \beta_{4} + 67 \beta_{5} - 135 \beta_{6} - 175 \beta_{8} + 64 \beta_{9} + 67 \beta_{10} - 17 \beta_{11} - 21 \beta_{12} + 41 \beta_{13} + 41 \beta_{15} ) q^{81} + ( -4229 - 44 \beta_{1} + 124 \beta_{2} + 1325 \beta_{3} - 4157 \beta_{4} + 230 \beta_{5} + 441 \beta_{6} - 44 \beta_{7} + 383 \beta_{8} + \beta_{9} - 52 \beta_{11} + 124 \beta_{12} - \beta_{14} ) q^{83} + ( 257 + 18 \beta_{1} + 520 \beta_{2} + 1165 \beta_{3} + 13328 \beta_{4} + 10 \beta_{5} + 161 \beta_{6} - 32 \beta_{7} + 224 \beta_{8} - 12 \beta_{9} - 181 \beta_{10} - 35 \beta_{11} + 2 \beta_{12} - 35 \beta_{13} + 60 \beta_{14} - 2 \beta_{15} ) q^{85} + ( 9220 - 26 \beta_{1} - 2492 \beta_{2} + 94 \beta_{3} - 9046 \beta_{4} + 390 \beta_{6} + 26 \beta_{7} + 426 \beta_{8} + 14 \beta_{9} - 120 \beta_{10} + 94 \beta_{11} - 80 \beta_{12} - 70 \beta_{13} + 14 \beta_{14} ) q^{87} + ( -48 - 496 \beta_{2} - 602 \beta_{3} - 7226 \beta_{4} + 190 \beta_{5} + 66 \beta_{6} - 14 \beta_{7} - 2 \beta_{8} - 190 \beta_{10} - 58 \beta_{11} + 48 \beta_{12} + 6 \beta_{13} - 80 \beta_{14} - 6 \beta_{15} ) q^{89} + ( -205 + 1542 \beta_{2} + 1260 \beta_{3} + 22143 \beta_{4} - 25 \beta_{5} + 349 \beta_{6} - 22 \beta_{7} - 845 \beta_{8} + 25 \beta_{10} - 77 \beta_{11} + 205 \beta_{12} + 29 \beta_{13} - 80 \beta_{14} - 29 \beta_{15} ) q^{91} + ( 6417 - 36 \beta_{1} - 2977 \beta_{2} - 55 \beta_{3} - 6479 \beta_{4} - 16 \beta_{6} + 36 \beta_{7} - 584 \beta_{8} - 76 \beta_{9} + 352 \beta_{10} - 55 \beta_{11} + 7 \beta_{12} - 40 \beta_{13} - 76 \beta_{14} ) q^{93} + ( 17293 + 25 \beta_{1} + 275 \beta_{2} - 1700 \beta_{3} - 16446 \beta_{4} - 110 \beta_{5} - 35 \beta_{6} + 75 \beta_{7} - 379 \beta_{8} + 80 \beta_{9} + 195 \beta_{10} + 125 \beta_{11} - 50 \beta_{12} + 35 \beta_{13} + 15 \beta_{14} - 30 \beta_{15} ) q^{95} + ( -676 + 86 \beta_{1} - 18 \beta_{2} - 521 \beta_{3} - 781 \beta_{4} - 21 \beta_{5} + 83 \beta_{6} + 86 \beta_{7} + 482 \beta_{8} - 64 \beta_{9} - 87 \beta_{11} - 18 \beta_{12} + 64 \beta_{14} + \beta_{15} ) q^{97} + ( 8707 - 190 \beta_{1} + 1717 \beta_{2} - 1491 \beta_{3} - 51 \beta_{4} - 87 \beta_{5} - 117 \beta_{6} + 1706 \beta_{8} - 95 \beta_{9} - 87 \beta_{10} - 51 \beta_{11} + 277 \beta_{12} - 69 \beta_{13} - 69 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 10q^{3} - 42q^{5} - 86q^{7} + O(q^{10}) \) \( 16q + 10q^{3} - 42q^{5} - 86q^{7} + 536q^{13} - 698q^{15} - 1828q^{17} + 2512q^{19} - 4284q^{21} - 7642q^{23} + 9140q^{25} - 12272q^{27} + 11876q^{33} + 10518q^{35} - 7620q^{37} + 11244q^{39} - 21284q^{41} + 20002q^{43} + 686q^{45} + 25298q^{47} + 12852q^{53} - 10584q^{55} + 55848q^{57} - 142704q^{59} - 20564q^{61} - 115282q^{63} - 38256q^{65} - 10506q^{67} + 15432q^{73} + 256226q^{75} + 133852q^{77} - 159344q^{79} - 236116q^{81} - 61222q^{83} + 7056q^{85} + 162176q^{87} + 122180q^{93} + 267512q^{95} - 17344q^{97} + 107332q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 1375 x^{14} + 743087 x^{12} + 198706725 x^{10} + 26872635188 x^{8} + 1612811892960 x^{6} + 24971013117824 x^{4} + 124211190826240 x^{2} + 177426662425600\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{2} + 344 \)
\(\beta_{2}\)\(=\)\((\)\(-11825110207571 \nu^{14} - 13462238412424545 \nu^{12} - 5682876639346407577 \nu^{10} - 1095404202095902328315 \nu^{8} - 97152181490387236940048 \nu^{6} - 3863833615055145620098720 \nu^{4} - 90565141677532700310107904 \nu^{2} + 250967095756936052654361600 \nu + 192797603188806497285044480\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(11825110207571 \nu^{14} + 13462238412424545 \nu^{12} + 5682876639346407577 \nu^{10} + 1095404202095902328315 \nu^{8} + 97152181490387236940048 \nu^{6} + 3863833615055145620098720 \nu^{4} + 90565141677532700310107904 \nu^{2} + 250967095756936052654361600 \nu - 192797603188806497285044480\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-1809265083797853191 \nu^{15} - 2507428535219858004045 \nu^{13} - 1366856259525550450056517 \nu^{11} - 368975242720366525158266015 \nu^{9} - 50443590459859677634015726208 \nu^{7} - 3079764569891573188210805156320 \nu^{5} - 51612542386876249198327508150784 \nu^{3} - 375523742774793743806881372849920 \nu\)\()/ \)\(41\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(6866491718161395654215 \nu^{15} - 357160040187741313248744 \nu^{14} + 11069663104785560762888205 \nu^{13} - 482670369734028581607211512 \nu^{12} + 7181471713121377381025820805 \nu^{11} - 255630176344657651316440839480 \nu^{10} + 2382683421778575594622394461535 \nu^{9} - 66846944600066252228480518367016 \nu^{8} + 422523400440708528419984044413440 \nu^{7} - 8847276528455532184796971776315648 \nu^{6} + 37621430170979065387980115262923360 \nu^{5} - 525059848586542496097907220140069632 \nu^{4} + 1337439695782314706831377434958435840 \nu^{3} - 8575998357382048756480267774157488128 \nu^{2} + 7252751816283467948283657920432480000 \nu - 28406264753768725158262487417131223040\)\()/ \)\(76\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(20\!\cdots\!62\)\( \nu^{15} - \)\(74\!\cdots\!35\)\( \nu^{14} - \)\(28\!\cdots\!90\)\( \nu^{13} - \)\(10\!\cdots\!85\)\( \nu^{12} - \)\(15\!\cdots\!94\)\( \nu^{11} - \)\(56\!\cdots\!05\)\( \nu^{10} - \)\(40\!\cdots\!30\)\( \nu^{9} - \)\(15\!\cdots\!55\)\( \nu^{8} - \)\(53\!\cdots\!56\)\( \nu^{7} - \)\(20\!\cdots\!60\)\( \nu^{6} - \)\(29\!\cdots\!40\)\( \nu^{5} - \)\(12\!\cdots\!60\)\( \nu^{4} - \)\(30\!\cdots\!88\)\( \nu^{3} - \)\(16\!\cdots\!00\)\( \nu^{2} - \)\(47\!\cdots\!40\)\( \nu - \)\(46\!\cdots\!00\)\(\)\()/ \)\(22\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-13786776642126692303 \nu^{15} - 18903858510802945525485 \nu^{13} - 10190057495851634706399061 \nu^{11} - 2720317152186531734246366495 \nu^{9} - 368252411846707879595859317864 \nu^{7} - 22339702761128737788859802402560 \nu^{5} - 376172919280776681615944469343872 \nu^{3} - 2677877995500129055861418807087360 \nu\)\()/ \)\(87\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(55\!\cdots\!67\)\( \nu^{15} + \)\(69\!\cdots\!00\)\( \nu^{14} - \)\(76\!\cdots\!05\)\( \nu^{13} + \)\(94\!\cdots\!00\)\( \nu^{12} - \)\(40\!\cdots\!69\)\( \nu^{11} + \)\(50\!\cdots\!00\)\( \nu^{10} - \)\(10\!\cdots\!75\)\( \nu^{9} + \)\(13\!\cdots\!00\)\( \nu^{8} - \)\(14\!\cdots\!16\)\( \nu^{7} + \)\(17\!\cdots\!00\)\( \nu^{6} - \)\(84\!\cdots\!80\)\( \nu^{5} + \)\(95\!\cdots\!00\)\( \nu^{4} - \)\(11\!\cdots\!48\)\( \nu^{3} + \)\(96\!\cdots\!00\)\( \nu^{2} - \)\(32\!\cdots\!40\)\( \nu + \)\(16\!\cdots\!00\)\(\)\()/ \)\(26\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(55\!\cdots\!67\)\( \nu^{15} + \)\(12\!\cdots\!40\)\( \nu^{14} - \)\(76\!\cdots\!05\)\( \nu^{13} + \)\(17\!\cdots\!40\)\( \nu^{12} - \)\(40\!\cdots\!69\)\( \nu^{11} + \)\(94\!\cdots\!20\)\( \nu^{10} - \)\(10\!\cdots\!75\)\( \nu^{9} + \)\(24\!\cdots\!20\)\( \nu^{8} - \)\(14\!\cdots\!16\)\( \nu^{7} + \)\(32\!\cdots\!40\)\( \nu^{6} - \)\(84\!\cdots\!80\)\( \nu^{5} + \)\(18\!\cdots\!40\)\( \nu^{4} - \)\(11\!\cdots\!48\)\( \nu^{3} + \)\(20\!\cdots\!00\)\( \nu^{2} - \)\(32\!\cdots\!40\)\( \nu + \)\(42\!\cdots\!00\)\(\)\()/ \)\(26\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(20\!\cdots\!59\)\( \nu^{15} + \)\(83\!\cdots\!45\)\( \nu^{14} - \)\(28\!\cdots\!65\)\( \nu^{13} + \)\(12\!\cdots\!75\)\( \nu^{12} - \)\(15\!\cdots\!93\)\( \nu^{11} + \)\(71\!\cdots\!15\)\( \nu^{10} - \)\(41\!\cdots\!15\)\( \nu^{9} + \)\(20\!\cdots\!25\)\( \nu^{8} - \)\(55\!\cdots\!72\)\( \nu^{7} + \)\(28\!\cdots\!60\)\( \nu^{6} - \)\(33\!\cdots\!40\)\( \nu^{5} + \)\(15\!\cdots\!00\)\( \nu^{4} - \)\(48\!\cdots\!76\)\( \nu^{3} - \)\(13\!\cdots\!20\)\( \nu^{2} - \)\(15\!\cdots\!80\)\( \nu - \)\(69\!\cdots\!00\)\(\)\()/ \)\(66\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(10\!\cdots\!03\)\( \nu^{15} - \)\(53\!\cdots\!60\)\( \nu^{14} + \)\(13\!\cdots\!85\)\( \nu^{13} - \)\(74\!\cdots\!20\)\( \nu^{12} + \)\(73\!\cdots\!61\)\( \nu^{11} - \)\(40\!\cdots\!40\)\( \nu^{10} + \)\(19\!\cdots\!95\)\( \nu^{9} - \)\(11\!\cdots\!60\)\( \nu^{8} + \)\(25\!\cdots\!64\)\( \nu^{7} - \)\(15\!\cdots\!40\)\( \nu^{6} + \)\(14\!\cdots\!60\)\( \nu^{5} - \)\(89\!\cdots\!20\)\( \nu^{4} + \)\(14\!\cdots\!72\)\( \nu^{3} - \)\(11\!\cdots\!60\)\( \nu^{2} + \)\(22\!\cdots\!60\)\( \nu - \)\(33\!\cdots\!00\)\(\)\()/ \)\(26\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(16\!\cdots\!01\)\( \nu^{15} + \)\(13\!\cdots\!70\)\( \nu^{14} + \)\(22\!\cdots\!15\)\( \nu^{13} + \)\(18\!\cdots\!50\)\( \nu^{12} + \)\(12\!\cdots\!07\)\( \nu^{11} + \)\(10\!\cdots\!90\)\( \nu^{10} + \)\(32\!\cdots\!25\)\( \nu^{9} + \)\(26\!\cdots\!50\)\( \nu^{8} + \)\(43\!\cdots\!48\)\( \nu^{7} + \)\(34\!\cdots\!60\)\( \nu^{6} + \)\(25\!\cdots\!40\)\( \nu^{5} + \)\(19\!\cdots\!00\)\( \nu^{4} + \)\(33\!\cdots\!44\)\( \nu^{3} + \)\(19\!\cdots\!80\)\( \nu^{2} + \)\(95\!\cdots\!20\)\( \nu + \)\(32\!\cdots\!00\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(10\!\cdots\!43\)\( \nu^{15} - \)\(27\!\cdots\!25\)\( \nu^{14} - \)\(15\!\cdots\!65\)\( \nu^{13} - \)\(37\!\cdots\!75\)\( \nu^{12} - \)\(80\!\cdots\!21\)\( \nu^{11} - \)\(20\!\cdots\!75\)\( \nu^{10} - \)\(21\!\cdots\!35\)\( \nu^{9} - \)\(53\!\cdots\!25\)\( \nu^{8} - \)\(28\!\cdots\!24\)\( \nu^{7} - \)\(70\!\cdots\!00\)\( \nu^{6} - \)\(16\!\cdots\!40\)\( \nu^{5} - \)\(40\!\cdots\!00\)\( \nu^{4} - \)\(20\!\cdots\!12\)\( \nu^{3} - \)\(47\!\cdots\!00\)\( \nu^{2} - \)\(58\!\cdots\!60\)\( \nu - \)\(13\!\cdots\!00\)\(\)\()/ \)\(66\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(13\!\cdots\!84\)\( \nu^{15} + \)\(22\!\cdots\!05\)\( \nu^{14} + \)\(17\!\cdots\!00\)\( \nu^{13} + \)\(30\!\cdots\!55\)\( \nu^{12} + \)\(96\!\cdots\!28\)\( \nu^{11} + \)\(16\!\cdots\!15\)\( \nu^{10} + \)\(25\!\cdots\!20\)\( \nu^{9} + \)\(45\!\cdots\!65\)\( \nu^{8} + \)\(34\!\cdots\!52\)\( \nu^{7} + \)\(62\!\cdots\!80\)\( \nu^{6} + \)\(19\!\cdots\!00\)\( \nu^{5} + \)\(37\!\cdots\!80\)\( \nu^{4} + \)\(22\!\cdots\!36\)\( \nu^{3} + \)\(49\!\cdots\!00\)\( \nu^{2} + \)\(48\!\cdots\!80\)\( \nu + \)\(13\!\cdots\!00\)\(\)\()/ \)\(66\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(27\!\cdots\!53\)\( \nu^{15} - \)\(46\!\cdots\!30\)\( \nu^{14} + \)\(37\!\cdots\!35\)\( \nu^{13} - \)\(63\!\cdots\!30\)\( \nu^{12} + \)\(20\!\cdots\!11\)\( \nu^{11} - \)\(33\!\cdots\!90\)\( \nu^{10} + \)\(53\!\cdots\!45\)\( \nu^{9} - \)\(88\!\cdots\!90\)\( \nu^{8} + \)\(71\!\cdots\!64\)\( \nu^{7} - \)\(11\!\cdots\!80\)\( \nu^{6} + \)\(41\!\cdots\!60\)\( \nu^{5} - \)\(65\!\cdots\!80\)\( \nu^{4} + \)\(51\!\cdots\!72\)\( \nu^{3} - \)\(75\!\cdots\!00\)\( \nu^{2} + \)\(14\!\cdots\!60\)\( \nu - \)\(20\!\cdots\!00\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{1} - 344\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - 3 \beta_{11} + 4 \beta_{10} - 3 \beta_{7} + 25 \beta_{6} - 4 \beta_{5} + 269 \beta_{4} - 290 \beta_{3} - 288 \beta_{2} + 1\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-49 \beta_{15} - 49 \beta_{13} - 3 \beta_{12} + 25 \beta_{11} - 35 \beta_{10} - 56 \beta_{9} - 25 \beta_{8} + 135 \beta_{6} - 35 \beta_{5} + 25 \beta_{4} - 1045 \beta_{3} + 1067 \beta_{2} - 712 \beta_{1} + 203177\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-423 \beta_{15} - 663 \beta_{14} + 423 \beta_{13} + 321 \beta_{12} + 1797 \beta_{11} - 2202 \beta_{10} + 918 \beta_{8} + 1357 \beta_{7} - 14493 \beta_{6} + 2202 \beta_{5} - 277011 \beta_{4} + 92938 \beta_{3} + 91462 \beta_{2} - 321\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(28051 \beta_{15} + 28051 \beta_{13} - 4959 \beta_{12} - 14515 \beta_{11} + 20789 \beta_{10} + 35684 \beta_{9} - 30125 \beta_{8} - 78849 \beta_{6} + 20789 \beta_{5} - 14515 \beta_{4} + 619267 \beta_{3} - 638741 \beta_{2} + 256244 \beta_{1} - 65791835\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(144878 \beta_{15} + 328013 \beta_{14} - 144878 \beta_{13} - 85532 \beta_{12} - 817110 \beta_{11} + 991451 \beta_{10} - 649323 \beta_{8} - 600379 \beta_{7} + 6511880 \beta_{6} - 991451 \beta_{5} + 144189820 \beta_{4} - 31628703 \beta_{3} - 30897125 \beta_{2} + 85532\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-12331845 \beta_{15} - 12331845 \beta_{13} + 4799217 \beta_{12} + 6328509 \beta_{11} - 10704279 \beta_{10} - 17257200 \beta_{9} + 32093091 \beta_{8} + 36018315 \beta_{6} - 10704279 \beta_{5} + 6328509 \beta_{4} - 291744033 \beta_{3} + 302871759 \beta_{2} - 93830784 \beta_{1} + 22568502013\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-44989245 \beta_{15} - 146692755 \beta_{14} + 44989245 \beta_{13} + 19862799 \beta_{12} + 337870959 \beta_{11} - 412227204 \beta_{10} + 332776008 \beta_{8} + 263231913 \beta_{7} - 2676124203 \beta_{6} + 412227204 \beta_{5} - 65091322317 \beta_{4} + 11208592876 \beta_{3} + 10890584716 \beta_{2} - 19862799\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(4950034887 \beta_{15} + 4950034887 \beta_{13} - 2926608051 \beta_{12} - 2531335191 \beta_{11} + 5135564073 \beta_{10} + 7524605628 \beta_{9} - 20319748233 \beta_{8} - 15260904837 \beta_{6} + 5135564073 \beta_{5} - 2531335191 \beta_{4} + 128447611983 \beta_{3} - 133905555225 \beta_{2} + 34781451596 \beta_{1} - 8047494130639\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(12950771260 \beta_{15} + 62493690265 \beta_{14} - 12950771260 \beta_{13} - 3616179250 \beta_{12} - 134053666536 \beta_{11} + 164735847133 \beta_{10} - 150271130133 \beta_{8} - 114159684135 \beta_{7} + 1057453079134 \beta_{6} - 164735847133 \beta_{5} + 28018936529822 \beta_{4} - 4078754461145 \beta_{3} - 3948316973859 \beta_{2} + 3616179250\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(-1909566697801 \beta_{15} - 1909566697801 \beta_{13} + 1502887482645 \beta_{12} + 979767841969 \beta_{11} - 2349486909563 \beta_{10} - 3121936989704 \beta_{9} + 10669615399535 \beta_{8} + 6268558277439 \beta_{6} - 2349486909563 \beta_{5} + 979767841969 \beta_{4} - 55063108048765 \beta_{3} + 57545763373379 \beta_{2} - 13007333386936 \beta_{1} + 2943173670069761\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(-3397983388347 \beta_{15} - 25911317704287 \beta_{14} + 3397983388347 \beta_{13} + 180343826229 \beta_{12} + 52152774111345 \beta_{11} - 64481759886486 \beta_{10} + 63760384581570 \beta_{8} + 48932261054389 \beta_{7} - 410105689035537 \beta_{6} + 64481759886486 \beta_{5} - 11822677651707999 \beta_{4} + 1510156029970678 \beta_{3} + 1458183599685562 \beta_{2} - 180343826229\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(722959127959243 \beta_{15} + 722959127959243 \beta_{13} - 707341430714391 \beta_{12} - 374060778641803 \beta_{11} + 1038943847453837 \beta_{10} + 1261726943170772 \beta_{9} - 5098801118992469 \beta_{8} - 2535186962021049 \beta_{6} + 1038943847453837 \beta_{5} - 374060778641803 \beta_{4} + 23278843914380203 \beta_{3} - 24360246123736397 \beta_{2} + 4896953527466948 \beta_{1} - 1094329166348265875\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(752369968134842 \beta_{15} + 10568378652706277 \beta_{14} - 752369968134842 \beta_{13} + 298058147893480 \beta_{12} - 20099220017949066 \beta_{11} + 24970170033860015 \beta_{10} - 26162402625433935 \beta_{8} - 20737402517525971 \beta_{7} + 157638608730530372 \beta_{6} - 24970170033860015 \beta_{5} + 4934989342780262800 \beta_{4} - 565644165234909891 \beta_{3} - 545246887069067345 \beta_{2} - 298058147893480\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(\beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
18.6675i
14.0500i
13.0932i
1.56776i
2.29308i
3.47866i
15.6412i
19.8299i
18.6675i
14.0500i
13.0932i
1.56776i
2.29308i
3.47866i
15.6412i
19.8299i
0 −17.6675 + 17.6675i 0 55.7953 3.44692i 0 52.6478 + 52.6478i 0 381.282i 0
63.2 0 −13.0500 + 13.0500i 0 −30.1056 47.1026i 0 −150.919 150.919i 0 97.6048i 0
63.3 0 −12.0932 + 12.0932i 0 −34.4070 + 44.0586i 0 74.8536 + 74.8536i 0 49.4886i 0
63.4 0 2.56776 2.56776i 0 3.91778 + 55.7642i 0 −86.5899 86.5899i 0 229.813i 0
63.5 0 3.29308 3.29308i 0 35.3847 43.2773i 0 −5.96093 5.96093i 0 221.311i 0
63.6 0 4.47866 4.47866i 0 −49.0494 26.8170i 0 132.222 + 132.222i 0 202.883i 0
63.7 0 16.6412 16.6412i 0 53.0700 + 17.5663i 0 76.6521 + 76.6521i 0 310.861i 0
63.8 0 20.8299 20.8299i 0 −55.6057 5.74531i 0 −135.906 135.906i 0 624.772i 0
127.1 0 −17.6675 17.6675i 0 55.7953 + 3.44692i 0 52.6478 52.6478i 0 381.282i 0
127.2 0 −13.0500 13.0500i 0 −30.1056 + 47.1026i 0 −150.919 + 150.919i 0 97.6048i 0
127.3 0 −12.0932 12.0932i 0 −34.4070 44.0586i 0 74.8536 74.8536i 0 49.4886i 0
127.4 0 2.56776 + 2.56776i 0 3.91778 55.7642i 0 −86.5899 + 86.5899i 0 229.813i 0
127.5 0 3.29308 + 3.29308i 0 35.3847 + 43.2773i 0 −5.96093 + 5.96093i 0 221.311i 0
127.6 0 4.47866 + 4.47866i 0 −49.0494 + 26.8170i 0 132.222 132.222i 0 202.883i 0
127.7 0 16.6412 + 16.6412i 0 53.0700 17.5663i 0 76.6521 76.6521i 0 310.861i 0
127.8 0 20.8299 + 20.8299i 0 −55.6057 + 5.74531i 0 −135.906 + 135.906i 0 624.772i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.6.n.d yes 16
4.b odd 2 1 160.6.n.c 16
5.c odd 4 1 160.6.n.c 16
20.e even 4 1 inner 160.6.n.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.n.c 16 4.b odd 2 1
160.6.n.c 16 5.c odd 4 1
160.6.n.d yes 16 1.a even 1 1 trivial
160.6.n.d yes 16 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{16} - \cdots\) acting on \(S_{6}^{\mathrm{new}}(160, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 10 T + 50 T^{2} + 3114 T^{3} - 71776 T^{4} + 1530706 T^{5} - 6869762 T^{6} + 282516126 T^{7} - 1414085636 T^{8} - 89177429938 T^{9} + 2520680189450 T^{10} - 49929440504142 T^{11} + 506618123321952 T^{12} + 709766597096154 T^{13} - 113295686701585050 T^{14} + 1816344389104634358 T^{15} - 43957043443472192442 T^{16} + \)\(44\!\cdots\!94\)\( T^{17} - \)\(66\!\cdots\!50\)\( T^{18} + \)\(10\!\cdots\!78\)\( T^{19} + \)\(17\!\cdots\!52\)\( T^{20} - \)\(42\!\cdots\!06\)\( T^{21} + \)\(51\!\cdots\!50\)\( T^{22} - \)\(44\!\cdots\!66\)\( T^{23} - \)\(17\!\cdots\!36\)\( T^{24} + \)\(83\!\cdots\!18\)\( T^{25} - \)\(49\!\cdots\!38\)\( T^{26} + \)\(26\!\cdots\!42\)\( T^{27} - \)\(30\!\cdots\!76\)\( T^{28} + \)\(32\!\cdots\!02\)\( T^{29} + \)\(12\!\cdots\!50\)\( T^{30} - \)\(60\!\cdots\!70\)\( T^{31} + \)\(14\!\cdots\!01\)\( T^{32} \)
$5$ \( 1 + 42 T - 3688 T^{2} - 336370 T^{3} - 1345940 T^{4} + 732464050 T^{5} + 13329225000 T^{6} - 337978756250 T^{7} - 9350412656250 T^{8} - 1056183613281250 T^{9} + 130168212890625000 T^{10} + 22353028869628906250 T^{11} - \)\(12\!\cdots\!00\)\( T^{12} - \)\(10\!\cdots\!50\)\( T^{13} - \)\(34\!\cdots\!00\)\( T^{14} + \)\(12\!\cdots\!50\)\( T^{15} + \)\(90\!\cdots\!25\)\( T^{16} \)
$7$ \( 1 + 86 T + 3698 T^{2} - 2808286 T^{3} - 810527584 T^{4} + 2504717314 T^{5} + 7155971823534 T^{6} + 2324030762443574 T^{7} + 166847822527644348 T^{8} - 29386477550704152562 T^{9} - \)\(38\!\cdots\!46\)\( T^{10} - \)\(51\!\cdots\!62\)\( T^{11} + \)\(38\!\cdots\!24\)\( T^{12} + \)\(97\!\cdots\!26\)\( T^{13} + \)\(45\!\cdots\!90\)\( T^{14} - \)\(35\!\cdots\!26\)\( T^{15} - \)\(24\!\cdots\!62\)\( T^{16} - \)\(60\!\cdots\!82\)\( T^{17} + \)\(12\!\cdots\!10\)\( T^{18} + \)\(46\!\cdots\!18\)\( T^{19} + \)\(30\!\cdots\!24\)\( T^{20} - \)\(69\!\cdots\!34\)\( T^{21} - \)\(87\!\cdots\!54\)\( T^{22} - \)\(11\!\cdots\!66\)\( T^{23} + \)\(10\!\cdots\!48\)\( T^{24} + \)\(24\!\cdots\!18\)\( T^{25} + \)\(12\!\cdots\!66\)\( T^{26} + \)\(75\!\cdots\!02\)\( T^{27} - \)\(41\!\cdots\!84\)\( T^{28} - \)\(23\!\cdots\!02\)\( T^{29} + \)\(53\!\cdots\!02\)\( T^{30} + \)\(20\!\cdots\!98\)\( T^{31} + \)\(40\!\cdots\!01\)\( T^{32} \)
$11$ \( 1 - 580860 T^{2} + 233332093584 T^{4} - 74638580678134772 T^{6} + \)\(20\!\cdots\!20\)\( T^{8} - \)\(46\!\cdots\!24\)\( T^{10} + \)\(96\!\cdots\!08\)\( T^{12} - \)\(18\!\cdots\!84\)\( T^{14} + \)\(30\!\cdots\!54\)\( T^{16} - \)\(46\!\cdots\!84\)\( T^{18} + \)\(65\!\cdots\!08\)\( T^{20} - \)\(81\!\cdots\!24\)\( T^{22} + \)\(90\!\cdots\!20\)\( T^{24} - \)\(87\!\cdots\!72\)\( T^{26} + \)\(71\!\cdots\!84\)\( T^{28} - \)\(45\!\cdots\!60\)\( T^{30} + \)\(20\!\cdots\!01\)\( T^{32} \)
$13$ \( 1 - 536 T + 143648 T^{2} - 293170824 T^{3} - 185935117448 T^{4} + 112653613917608 T^{5} + 9301436713751904 T^{6} + 69204174166800470584 T^{7} + \)\(13\!\cdots\!60\)\( T^{8} - \)\(25\!\cdots\!64\)\( T^{9} - \)\(50\!\cdots\!68\)\( T^{10} - \)\(90\!\cdots\!08\)\( T^{11} + \)\(40\!\cdots\!76\)\( T^{12} + \)\(23\!\cdots\!68\)\( T^{13} + \)\(13\!\cdots\!08\)\( T^{14} + \)\(14\!\cdots\!44\)\( T^{15} - \)\(96\!\cdots\!10\)\( T^{16} + \)\(55\!\cdots\!92\)\( T^{17} + \)\(17\!\cdots\!92\)\( T^{18} + \)\(12\!\cdots\!76\)\( T^{19} + \)\(77\!\cdots\!76\)\( T^{20} - \)\(64\!\cdots\!44\)\( T^{21} - \)\(13\!\cdots\!32\)\( T^{22} - \)\(24\!\cdots\!48\)\( T^{23} + \)\(48\!\cdots\!60\)\( T^{24} + \)\(92\!\cdots\!12\)\( T^{25} + \)\(46\!\cdots\!96\)\( T^{26} + \)\(20\!\cdots\!56\)\( T^{27} - \)\(12\!\cdots\!48\)\( T^{28} - \)\(74\!\cdots\!32\)\( T^{29} + \)\(13\!\cdots\!52\)\( T^{30} - \)\(18\!\cdots\!52\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \)
$17$ \( 1 + 1828 T + 1670792 T^{2} - 327758324 T^{3} + 3015753425928 T^{4} + 10493189333351412 T^{5} + 14196566162828930648 T^{6} + \)\(64\!\cdots\!44\)\( T^{7} + \)\(60\!\cdots\!96\)\( T^{8} + \)\(26\!\cdots\!20\)\( T^{9} + \)\(48\!\cdots\!00\)\( T^{10} + \)\(40\!\cdots\!40\)\( T^{11} + \)\(30\!\cdots\!88\)\( T^{12} + \)\(53\!\cdots\!44\)\( T^{13} + \)\(10\!\cdots\!76\)\( T^{14} + \)\(94\!\cdots\!68\)\( T^{15} + \)\(94\!\cdots\!74\)\( T^{16} + \)\(13\!\cdots\!76\)\( T^{17} + \)\(20\!\cdots\!24\)\( T^{18} + \)\(15\!\cdots\!92\)\( T^{19} + \)\(12\!\cdots\!88\)\( T^{20} + \)\(23\!\cdots\!80\)\( T^{21} + \)\(39\!\cdots\!00\)\( T^{22} + \)\(30\!\cdots\!60\)\( T^{23} + \)\(99\!\cdots\!96\)\( T^{24} + \)\(15\!\cdots\!08\)\( T^{25} + \)\(47\!\cdots\!52\)\( T^{26} + \)\(49\!\cdots\!16\)\( T^{27} + \)\(20\!\cdots\!28\)\( T^{28} - \)\(31\!\cdots\!68\)\( T^{29} + \)\(22\!\cdots\!08\)\( T^{30} + \)\(35\!\cdots\!04\)\( T^{31} + \)\(27\!\cdots\!01\)\( T^{32} \)
$19$ \( ( 1 - 1256 T + 11933496 T^{2} - 17157951112 T^{3} + 74779232061244 T^{4} - 104521619172501960 T^{5} + \)\(31\!\cdots\!88\)\( T^{6} - \)\(38\!\cdots\!64\)\( T^{7} + \)\(93\!\cdots\!18\)\( T^{8} - \)\(94\!\cdots\!36\)\( T^{9} + \)\(19\!\cdots\!88\)\( T^{10} - \)\(15\!\cdots\!40\)\( T^{11} + \)\(28\!\cdots\!44\)\( T^{12} - \)\(15\!\cdots\!88\)\( T^{13} + \)\(27\!\cdots\!96\)\( T^{14} - \)\(71\!\cdots\!44\)\( T^{15} + \)\(14\!\cdots\!01\)\( T^{16} )^{2} \)
$23$ \( 1 + 7642 T + 29200082 T^{2} + 114423930750 T^{3} + 421188911449792 T^{4} + 1070557300674963934 T^{5} + \)\(24\!\cdots\!34\)\( T^{6} + \)\(58\!\cdots\!94\)\( T^{7} + \)\(77\!\cdots\!36\)\( T^{8} - \)\(18\!\cdots\!62\)\( T^{9} - \)\(19\!\cdots\!06\)\( T^{10} - \)\(97\!\cdots\!06\)\( T^{11} - \)\(29\!\cdots\!68\)\( T^{12} - \)\(18\!\cdots\!70\)\( T^{13} + \)\(73\!\cdots\!74\)\( T^{14} + \)\(37\!\cdots\!54\)\( T^{15} + \)\(12\!\cdots\!78\)\( T^{16} + \)\(24\!\cdots\!22\)\( T^{17} + \)\(30\!\cdots\!26\)\( T^{18} - \)\(49\!\cdots\!90\)\( T^{19} - \)\(50\!\cdots\!68\)\( T^{20} - \)\(10\!\cdots\!58\)\( T^{21} - \)\(14\!\cdots\!94\)\( T^{22} - \)\(84\!\cdots\!34\)\( T^{23} + \)\(22\!\cdots\!36\)\( T^{24} + \)\(11\!\cdots\!42\)\( T^{25} + \)\(29\!\cdots\!66\)\( T^{26} + \)\(84\!\cdots\!38\)\( T^{27} + \)\(21\!\cdots\!92\)\( T^{28} + \)\(37\!\cdots\!50\)\( T^{29} + \)\(61\!\cdots\!18\)\( T^{30} + \)\(10\!\cdots\!94\)\( T^{31} + \)\(86\!\cdots\!01\)\( T^{32} \)
$29$ \( 1 - 129450464 T^{2} + 9016787664231832 T^{4} - \)\(44\!\cdots\!56\)\( T^{6} + \)\(16\!\cdots\!60\)\( T^{8} - \)\(53\!\cdots\!48\)\( T^{10} + \)\(14\!\cdots\!16\)\( T^{12} - \)\(34\!\cdots\!32\)\( T^{14} + \)\(74\!\cdots\!82\)\( T^{16} - \)\(14\!\cdots\!32\)\( T^{18} + \)\(25\!\cdots\!16\)\( T^{20} - \)\(39\!\cdots\!48\)\( T^{22} + \)\(52\!\cdots\!60\)\( T^{24} - \)\(58\!\cdots\!56\)\( T^{26} + \)\(49\!\cdots\!32\)\( T^{28} - \)\(30\!\cdots\!64\)\( T^{30} + \)\(98\!\cdots\!01\)\( T^{32} \)
$31$ \( 1 - 221015276 T^{2} + 25619057922480912 T^{4} - \)\(20\!\cdots\!04\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{8} - \)\(64\!\cdots\!72\)\( T^{10} + \)\(27\!\cdots\!96\)\( T^{12} - \)\(98\!\cdots\!48\)\( T^{14} + \)\(30\!\cdots\!82\)\( T^{16} - \)\(80\!\cdots\!48\)\( T^{18} + \)\(18\!\cdots\!96\)\( T^{20} - \)\(35\!\cdots\!72\)\( T^{22} + \)\(57\!\cdots\!00\)\( T^{24} - \)\(76\!\cdots\!04\)\( T^{26} + \)\(77\!\cdots\!12\)\( T^{28} - \)\(54\!\cdots\!76\)\( T^{30} + \)\(20\!\cdots\!01\)\( T^{32} \)
$37$ \( 1 + 7620 T + 29032200 T^{2} + 1965688293660 T^{3} + 22987009602091912 T^{4} + 46723925283632703348 T^{5} + \)\(16\!\cdots\!60\)\( T^{6} + \)\(27\!\cdots\!40\)\( T^{7} + \)\(98\!\cdots\!88\)\( T^{8} + \)\(66\!\cdots\!16\)\( T^{9} + \)\(19\!\cdots\!52\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{11} + \)\(55\!\cdots\!44\)\( T^{12} + \)\(97\!\cdots\!64\)\( T^{13} + \)\(11\!\cdots\!44\)\( T^{14} + \)\(70\!\cdots\!36\)\( T^{15} + \)\(48\!\cdots\!10\)\( T^{16} + \)\(48\!\cdots\!52\)\( T^{17} + \)\(53\!\cdots\!56\)\( T^{18} + \)\(32\!\cdots\!52\)\( T^{19} + \)\(12\!\cdots\!44\)\( T^{20} + \)\(21\!\cdots\!00\)\( T^{21} + \)\(22\!\cdots\!48\)\( T^{22} + \)\(51\!\cdots\!88\)\( T^{23} + \)\(52\!\cdots\!88\)\( T^{24} + \)\(10\!\cdots\!80\)\( T^{25} + \)\(41\!\cdots\!40\)\( T^{26} + \)\(83\!\cdots\!64\)\( T^{27} + \)\(28\!\cdots\!12\)\( T^{28} + \)\(16\!\cdots\!20\)\( T^{29} + \)\(17\!\cdots\!00\)\( T^{30} + \)\(31\!\cdots\!60\)\( T^{31} + \)\(28\!\cdots\!01\)\( T^{32} \)
$41$ \( ( 1 + 10642 T + 229001616 T^{2} + 2308310140798 T^{3} + 41488616565167196 T^{4} + \)\(33\!\cdots\!02\)\( T^{5} + \)\(55\!\cdots\!56\)\( T^{6} + \)\(38\!\cdots\!10\)\( T^{7} + \)\(61\!\cdots\!50\)\( T^{8} + \)\(44\!\cdots\!10\)\( T^{9} + \)\(74\!\cdots\!56\)\( T^{10} + \)\(52\!\cdots\!02\)\( T^{11} + \)\(74\!\cdots\!96\)\( T^{12} + \)\(48\!\cdots\!98\)\( T^{13} + \)\(55\!\cdots\!16\)\( T^{14} + \)\(29\!\cdots\!42\)\( T^{15} + \)\(32\!\cdots\!01\)\( T^{16} )^{2} \)
$43$ \( 1 - 20002 T + 200040002 T^{2} - 4643840984270 T^{3} + 57483685077584672 T^{4} + 83126825813520796778 T^{5} - \)\(23\!\cdots\!50\)\( T^{6} + \)\(76\!\cdots\!70\)\( T^{7} - \)\(32\!\cdots\!64\)\( T^{8} + \)\(35\!\cdots\!86\)\( T^{9} - \)\(28\!\cdots\!02\)\( T^{10} + \)\(51\!\cdots\!50\)\( T^{11} - \)\(19\!\cdots\!52\)\( T^{12} - \)\(54\!\cdots\!78\)\( T^{13} + \)\(53\!\cdots\!62\)\( T^{14} - \)\(11\!\cdots\!58\)\( T^{15} + \)\(24\!\cdots\!66\)\( T^{16} - \)\(17\!\cdots\!94\)\( T^{17} + \)\(11\!\cdots\!38\)\( T^{18} - \)\(17\!\cdots\!46\)\( T^{19} - \)\(89\!\cdots\!52\)\( T^{20} + \)\(35\!\cdots\!50\)\( T^{21} - \)\(28\!\cdots\!98\)\( T^{22} + \)\(53\!\cdots\!02\)\( T^{23} - \)\(70\!\cdots\!64\)\( T^{24} + \)\(24\!\cdots\!10\)\( T^{25} - \)\(11\!\cdots\!50\)\( T^{26} + \)\(57\!\cdots\!46\)\( T^{27} + \)\(58\!\cdots\!72\)\( T^{28} - \)\(69\!\cdots\!10\)\( T^{29} + \)\(44\!\cdots\!98\)\( T^{30} - \)\(64\!\cdots\!14\)\( T^{31} + \)\(47\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - 25298 T + 319994402 T^{2} - 6541896537590 T^{3} - 11741966210019840 T^{4} + \)\(87\!\cdots\!98\)\( T^{5} + \)\(31\!\cdots\!26\)\( T^{6} + \)\(28\!\cdots\!26\)\( T^{7} + \)\(89\!\cdots\!80\)\( T^{8} - \)\(49\!\cdots\!50\)\( T^{9} - \)\(48\!\cdots\!58\)\( T^{10} - \)\(14\!\cdots\!46\)\( T^{11} + \)\(91\!\cdots\!24\)\( T^{12} + \)\(38\!\cdots\!50\)\( T^{13} + \)\(47\!\cdots\!30\)\( T^{14} - \)\(60\!\cdots\!90\)\( T^{15} + \)\(93\!\cdots\!70\)\( T^{16} - \)\(13\!\cdots\!30\)\( T^{17} + \)\(25\!\cdots\!70\)\( T^{18} + \)\(45\!\cdots\!50\)\( T^{19} + \)\(25\!\cdots\!24\)\( T^{20} - \)\(94\!\cdots\!22\)\( T^{21} - \)\(70\!\cdots\!42\)\( T^{22} - \)\(16\!\cdots\!50\)\( T^{23} + \)\(68\!\cdots\!80\)\( T^{24} + \)\(49\!\cdots\!82\)\( T^{25} + \)\(12\!\cdots\!74\)\( T^{26} + \)\(80\!\cdots\!14\)\( T^{27} - \)\(24\!\cdots\!40\)\( T^{28} - \)\(31\!\cdots\!30\)\( T^{29} + \)\(35\!\cdots\!98\)\( T^{30} - \)\(64\!\cdots\!14\)\( T^{31} + \)\(58\!\cdots\!01\)\( T^{32} \)
$53$ \( 1 - 12852 T + 82586952 T^{2} + 12779415797620 T^{3} - 141875027604640888 T^{4} - \)\(55\!\cdots\!72\)\( T^{5} + \)\(16\!\cdots\!20\)\( T^{6} - \)\(16\!\cdots\!60\)\( T^{7} - \)\(86\!\cdots\!84\)\( T^{8} + \)\(14\!\cdots\!16\)\( T^{9} + \)\(64\!\cdots\!88\)\( T^{10} - \)\(74\!\cdots\!80\)\( T^{11} + \)\(67\!\cdots\!08\)\( T^{12} + \)\(29\!\cdots\!72\)\( T^{13} - \)\(53\!\cdots\!88\)\( T^{14} - \)\(64\!\cdots\!48\)\( T^{15} + \)\(25\!\cdots\!06\)\( T^{16} - \)\(27\!\cdots\!64\)\( T^{17} - \)\(94\!\cdots\!12\)\( T^{18} + \)\(21\!\cdots\!04\)\( T^{19} + \)\(20\!\cdots\!08\)\( T^{20} - \)\(95\!\cdots\!40\)\( T^{21} + \)\(34\!\cdots\!12\)\( T^{22} + \)\(32\!\cdots\!12\)\( T^{23} - \)\(80\!\cdots\!84\)\( T^{24} - \)\(64\!\cdots\!80\)\( T^{25} + \)\(27\!\cdots\!80\)\( T^{26} - \)\(38\!\cdots\!04\)\( T^{27} - \)\(40\!\cdots\!88\)\( T^{28} + \)\(15\!\cdots\!60\)\( T^{29} + \)\(41\!\cdots\!48\)\( T^{30} - \)\(26\!\cdots\!64\)\( T^{31} + \)\(87\!\cdots\!01\)\( T^{32} \)
$59$ \( ( 1 + 71352 T + 5147553272 T^{2} + 263083260595160 T^{3} + 11526051017122699388 T^{4} + \)\(44\!\cdots\!28\)\( T^{5} + \)\(15\!\cdots\!16\)\( T^{6} + \)\(46\!\cdots\!12\)\( T^{7} + \)\(13\!\cdots\!10\)\( T^{8} + \)\(32\!\cdots\!88\)\( T^{9} + \)\(77\!\cdots\!16\)\( T^{10} + \)\(16\!\cdots\!72\)\( T^{11} + \)\(30\!\cdots\!88\)\( T^{12} + \)\(49\!\cdots\!40\)\( T^{13} + \)\(68\!\cdots\!72\)\( T^{14} + \)\(68\!\cdots\!48\)\( T^{15} + \)\(68\!\cdots\!01\)\( T^{16} )^{2} \)
$61$ \( ( 1 + 10282 T + 3727559576 T^{2} + 11244934877598 T^{3} + 6582098174228731436 T^{4} - \)\(31\!\cdots\!30\)\( T^{5} + \)\(74\!\cdots\!72\)\( T^{6} - \)\(78\!\cdots\!62\)\( T^{7} + \)\(66\!\cdots\!54\)\( T^{8} - \)\(66\!\cdots\!62\)\( T^{9} + \)\(52\!\cdots\!72\)\( T^{10} - \)\(18\!\cdots\!30\)\( T^{11} + \)\(33\!\cdots\!36\)\( T^{12} + \)\(48\!\cdots\!98\)\( T^{13} + \)\(13\!\cdots\!76\)\( T^{14} + \)\(31\!\cdots\!82\)\( T^{15} + \)\(25\!\cdots\!01\)\( T^{16} )^{2} \)
$67$ \( 1 + 10506 T + 55188018 T^{2} + 64685094733110 T^{3} - 1145101582269918112 T^{4} - \)\(11\!\cdots\!66\)\( T^{5} + \)\(90\!\cdots\!70\)\( T^{6} - \)\(13\!\cdots\!94\)\( T^{7} - \)\(58\!\cdots\!16\)\( T^{8} + \)\(10\!\cdots\!70\)\( T^{9} + \)\(57\!\cdots\!86\)\( T^{10} - \)\(37\!\cdots\!82\)\( T^{11} + \)\(12\!\cdots\!56\)\( T^{12} + \)\(51\!\cdots\!34\)\( T^{13} + \)\(11\!\cdots\!26\)\( T^{14} + \)\(28\!\cdots\!74\)\( T^{15} - \)\(82\!\cdots\!58\)\( T^{16} + \)\(38\!\cdots\!18\)\( T^{17} + \)\(21\!\cdots\!74\)\( T^{18} + \)\(12\!\cdots\!62\)\( T^{19} + \)\(40\!\cdots\!56\)\( T^{20} - \)\(16\!\cdots\!74\)\( T^{21} + \)\(34\!\cdots\!14\)\( T^{22} + \)\(89\!\cdots\!10\)\( T^{23} - \)\(64\!\cdots\!16\)\( T^{24} - \)\(20\!\cdots\!58\)\( T^{25} + \)\(18\!\cdots\!30\)\( T^{26} - \)\(32\!\cdots\!38\)\( T^{27} - \)\(42\!\cdots\!12\)\( T^{28} + \)\(32\!\cdots\!70\)\( T^{29} + \)\(36\!\cdots\!82\)\( T^{30} + \)\(94\!\cdots\!58\)\( T^{31} + \)\(12\!\cdots\!01\)\( T^{32} \)
$71$ \( 1 - 13607740108 T^{2} + 94972263534983306896 T^{4} - \)\(45\!\cdots\!72\)\( T^{6} + \)\(16\!\cdots\!96\)\( T^{8} - \)\(50\!\cdots\!76\)\( T^{10} + \)\(12\!\cdots\!40\)\( T^{12} - \)\(27\!\cdots\!08\)\( T^{14} + \)\(53\!\cdots\!06\)\( T^{16} - \)\(90\!\cdots\!08\)\( T^{18} + \)\(13\!\cdots\!40\)\( T^{20} - \)\(17\!\cdots\!76\)\( T^{22} + \)\(18\!\cdots\!96\)\( T^{24} - \)\(16\!\cdots\!72\)\( T^{26} + \)\(11\!\cdots\!96\)\( T^{28} - \)\(52\!\cdots\!08\)\( T^{30} + \)\(12\!\cdots\!01\)\( T^{32} \)
$73$ \( 1 - 15432 T + 119073312 T^{2} - 56032493808632 T^{3} + 4235186228799589816 T^{4} + \)\(61\!\cdots\!96\)\( T^{5} + \)\(12\!\cdots\!48\)\( T^{6} - \)\(11\!\cdots\!24\)\( T^{7} - \)\(89\!\cdots\!60\)\( T^{8} + \)\(71\!\cdots\!68\)\( T^{9} - \)\(22\!\cdots\!48\)\( T^{10} + \)\(11\!\cdots\!04\)\( T^{11} - \)\(29\!\cdots\!68\)\( T^{12} - \)\(26\!\cdots\!56\)\( T^{13} + \)\(53\!\cdots\!44\)\( T^{14} + \)\(14\!\cdots\!88\)\( T^{15} + \)\(30\!\cdots\!30\)\( T^{16} + \)\(29\!\cdots\!84\)\( T^{17} + \)\(22\!\cdots\!56\)\( T^{18} - \)\(23\!\cdots\!92\)\( T^{19} - \)\(54\!\cdots\!68\)\( T^{20} + \)\(42\!\cdots\!72\)\( T^{21} - \)\(17\!\cdots\!52\)\( T^{22} + \)\(11\!\cdots\!76\)\( T^{23} - \)\(30\!\cdots\!60\)\( T^{24} - \)\(80\!\cdots\!32\)\( T^{25} + \)\(18\!\cdots\!52\)\( T^{26} + \)\(18\!\cdots\!72\)\( T^{27} + \)\(26\!\cdots\!16\)\( T^{28} - \)\(73\!\cdots\!76\)\( T^{29} + \)\(32\!\cdots\!88\)\( T^{30} - \)\(86\!\cdots\!24\)\( T^{31} + \)\(11\!\cdots\!01\)\( T^{32} \)
$79$ \( ( 1 + 79672 T + 15921600120 T^{2} + 883492472350200 T^{3} + \)\(11\!\cdots\!40\)\( T^{4} + \)\(53\!\cdots\!80\)\( T^{5} + \)\(56\!\cdots\!80\)\( T^{6} + \)\(23\!\cdots\!20\)\( T^{7} + \)\(20\!\cdots\!70\)\( T^{8} + \)\(71\!\cdots\!80\)\( T^{9} + \)\(53\!\cdots\!80\)\( T^{10} + \)\(15\!\cdots\!20\)\( T^{11} + \)\(10\!\cdots\!40\)\( T^{12} + \)\(24\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!20\)\( T^{14} + \)\(20\!\cdots\!28\)\( T^{15} + \)\(80\!\cdots\!01\)\( T^{16} )^{2} \)
$83$ \( 1 + 61222 T + 1874066642 T^{2} + 12255516798122 T^{3} - 16226769598202256704 T^{4} - \)\(11\!\cdots\!34\)\( T^{5} + \)\(23\!\cdots\!62\)\( T^{6} + \)\(43\!\cdots\!38\)\( T^{7} + \)\(11\!\cdots\!04\)\( T^{8} - \)\(19\!\cdots\!26\)\( T^{9} - \)\(94\!\cdots\!78\)\( T^{10} - \)\(75\!\cdots\!74\)\( T^{11} + \)\(22\!\cdots\!04\)\( T^{12} + \)\(26\!\cdots\!90\)\( T^{13} + \)\(45\!\cdots\!30\)\( T^{14} - \)\(37\!\cdots\!50\)\( T^{15} - \)\(11\!\cdots\!10\)\( T^{16} - \)\(14\!\cdots\!50\)\( T^{17} + \)\(70\!\cdots\!70\)\( T^{18} + \)\(16\!\cdots\!30\)\( T^{19} + \)\(53\!\cdots\!04\)\( T^{20} - \)\(72\!\cdots\!82\)\( T^{21} - \)\(35\!\cdots\!22\)\( T^{22} - \)\(27\!\cdots\!82\)\( T^{23} + \)\(68\!\cdots\!04\)\( T^{24} + \)\(98\!\cdots\!34\)\( T^{25} + \)\(21\!\cdots\!38\)\( T^{26} - \)\(39\!\cdots\!38\)\( T^{27} - \)\(22\!\cdots\!04\)\( T^{28} + \)\(67\!\cdots\!46\)\( T^{29} + \)\(40\!\cdots\!58\)\( T^{30} + \)\(52\!\cdots\!54\)\( T^{31} + \)\(33\!\cdots\!01\)\( T^{32} \)
$89$ \( 1 - 59849990032 T^{2} + \)\(17\!\cdots\!44\)\( T^{4} - \)\(33\!\cdots\!56\)\( T^{6} + \)\(45\!\cdots\!84\)\( T^{8} - \)\(48\!\cdots\!60\)\( T^{10} + \)\(42\!\cdots\!32\)\( T^{12} - \)\(30\!\cdots\!72\)\( T^{14} + \)\(18\!\cdots\!38\)\( T^{16} - \)\(94\!\cdots\!72\)\( T^{18} + \)\(40\!\cdots\!32\)\( T^{20} - \)\(14\!\cdots\!60\)\( T^{22} + \)\(43\!\cdots\!84\)\( T^{24} - \)\(97\!\cdots\!56\)\( T^{26} + \)\(16\!\cdots\!44\)\( T^{28} - \)\(17\!\cdots\!32\)\( T^{30} + \)\(89\!\cdots\!01\)\( T^{32} \)
$97$ \( 1 + 17344 T + 150407168 T^{2} - 246974906594432 T^{3} + 52594420471951685112 T^{4} + \)\(73\!\cdots\!08\)\( T^{5} + \)\(15\!\cdots\!48\)\( T^{6} + \)\(60\!\cdots\!32\)\( T^{7} + \)\(10\!\cdots\!36\)\( T^{8} + \)\(15\!\cdots\!56\)\( T^{9} + \)\(30\!\cdots\!48\)\( T^{10} - \)\(19\!\cdots\!76\)\( T^{11} + \)\(13\!\cdots\!92\)\( T^{12} + \)\(57\!\cdots\!52\)\( T^{13} + \)\(10\!\cdots\!20\)\( T^{14} + \)\(29\!\cdots\!16\)\( T^{15} + \)\(43\!\cdots\!78\)\( T^{16} + \)\(25\!\cdots\!12\)\( T^{17} + \)\(80\!\cdots\!80\)\( T^{18} + \)\(36\!\cdots\!36\)\( T^{19} + \)\(75\!\cdots\!92\)\( T^{20} - \)\(92\!\cdots\!32\)\( T^{21} + \)\(12\!\cdots\!52\)\( T^{22} + \)\(51\!\cdots\!08\)\( T^{23} + \)\(30\!\cdots\!36\)\( T^{24} + \)\(15\!\cdots\!24\)\( T^{25} + \)\(32\!\cdots\!52\)\( T^{26} + \)\(13\!\cdots\!44\)\( T^{27} + \)\(84\!\cdots\!12\)\( T^{28} - \)\(34\!\cdots\!24\)\( T^{29} + \)\(17\!\cdots\!32\)\( T^{30} + \)\(17\!\cdots\!92\)\( T^{31} + \)\(87\!\cdots\!01\)\( T^{32} \)
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