Properties

Label 160.6.d.a
Level 160
Weight 6
Character orbit 160.d
Analytic conductor 25.661
Analytic rank 0
Dimension 20
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.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} - 109104 x^{12} - 96128 x^{11} + 3580672 x^{10} - 1538048 x^{9} - 27930624 x^{8} + 79364096 x^{7} + 157024256 x^{6} - 926941184 x^{5} + 4244635648 x^{4} + 20937965568 x^{3} - 73014444032 x^{2} - 137438953472 x + 1099511627776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{90}\cdot 3^{4}\cdot 5^{12} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{2} q^{5} + ( 10 + \beta_{4} ) q^{7} + ( -81 - \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{2} q^{5} + ( 10 + \beta_{4} ) q^{7} + ( -81 - \beta_{4} + \beta_{5} ) q^{9} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{11} + ( 7 \beta_{1} + \beta_{2} - \beta_{9} ) q^{13} + ( -45 + \beta_{11} ) q^{15} -\beta_{7} q^{17} + ( 3 \beta_{1} + 12 \beta_{2} + \beta_{3} - \beta_{6} - \beta_{14} ) q^{19} + ( -44 \beta_{1} + 4 \beta_{2} + \beta_{3} - \beta_{9} + \beta_{10} ) q^{21} + ( 235 + 3 \beta_{4} + 4 \beta_{5} + \beta_{15} - \beta_{19} ) q^{23} -625 q^{25} + ( 69 \beta_{1} - 18 \beta_{2} + \beta_{3} + 2 \beta_{6} - \beta_{9} - \beta_{10} + \beta_{16} - \beta_{17} ) q^{27} + ( 53 \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{6} + \beta_{9} - \beta_{14} + \beta_{16} + 2 \beta_{17} - \beta_{18} ) q^{29} + ( -357 + 5 \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{11} - \beta_{19} ) q^{31} + ( 280 - 17 \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + \beta_{15} + 3 \beta_{19} ) q^{33} + ( -12 \beta_{1} + 10 \beta_{2} - \beta_{9} + \beta_{10} - \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{35} + ( 21 \beta_{1} - \beta_{2} - 4 \beta_{3} + 6 \beta_{6} - 6 \beta_{9} - \beta_{10} - 5 \beta_{14} - \beta_{16} + \beta_{18} ) q^{37} + ( 2239 - 25 \beta_{4} - 10 \beta_{5} + \beta_{7} + \beta_{8} + 4 \beta_{11} - 4 \beta_{12} + 4 \beta_{13} + 2 \beta_{15} + \beta_{19} ) q^{39} + ( 575 - 12 \beta_{4} - 18 \beta_{5} - 2 \beta_{7} + 10 \beta_{11} - \beta_{12} + 5 \beta_{13} + 3 \beta_{15} - 4 \beta_{19} ) q^{41} + ( 117 \beta_{1} - 104 \beta_{2} + 3 \beta_{3} + 11 \beta_{9} + 5 \beta_{10} + 3 \beta_{14} + 7 \beta_{16} + \beta_{17} + 3 \beta_{18} ) q^{43} + ( 68 \beta_{1} - 80 \beta_{2} + 4 \beta_{9} + \beta_{10} + 4 \beta_{14} + 4 \beta_{16} - \beta_{17} + \beta_{18} ) q^{45} + ( -2214 - 24 \beta_{5} - 6 \beta_{7} + 3 \beta_{11} + 2 \beta_{12} + 5 \beta_{13} + 4 \beta_{15} - 4 \beta_{19} ) q^{47} + ( 942 + 16 \beta_{4} + 18 \beta_{5} + 2 \beta_{8} - 10 \beta_{11} + 3 \beta_{12} - \beta_{13} + 5 \beta_{15} - 6 \beta_{19} ) q^{49} + ( 58 \beta_{1} + 142 \beta_{2} + 2 \beta_{3} + 17 \beta_{6} - 18 \beta_{9} - 6 \beta_{10} - 17 \beta_{14} - 6 \beta_{16} + 8 \beta_{17} - 4 \beta_{18} ) q^{51} + ( 250 \beta_{1} + 12 \beta_{2} + 16 \beta_{3} - 10 \beta_{6} + 5 \beta_{9} + 9 \beta_{10} - 3 \beta_{14} + 3 \beta_{16} - 6 \beta_{17} + 9 \beta_{18} ) q^{53} + ( 1209 - 5 \beta_{4} + 4 \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{11} - \beta_{13} + \beta_{15} + 4 \beta_{19} ) q^{55} + ( 260 + 24 \beta_{4} + 18 \beta_{5} + \beta_{7} + 4 \beta_{8} + 34 \beta_{11} - 6 \beta_{12} - 2 \beta_{15} - 4 \beta_{19} ) q^{57} + ( -231 \beta_{1} - 116 \beta_{2} - 3 \beta_{3} - \beta_{6} - 22 \beta_{9} + 6 \beta_{10} - 3 \beta_{14} + 6 \beta_{16} ) q^{59} + ( 153 \beta_{1} + 69 \beta_{2} + 22 \beta_{3} + 6 \beta_{6} - 14 \beta_{9} - 11 \beta_{10} - 7 \beta_{14} + 7 \beta_{16} - 9 \beta_{18} ) q^{61} + ( -12033 - 76 \beta_{4} + 60 \beta_{5} + 4 \beta_{7} + 2 \beta_{8} - 3 \beta_{11} - 10 \beta_{12} + 5 \beta_{13} + 5 \beta_{15} + 13 \beta_{19} ) q^{63} + ( 1 - 10 \beta_{4} + 11 \beta_{5} + \beta_{7} - \beta_{8} - 9 \beta_{11} - 5 \beta_{12} + \beta_{13} + 4 \beta_{15} + \beta_{19} ) q^{65} + ( -230 \beta_{1} + 314 \beta_{2} + 41 \beta_{3} + 6 \beta_{6} - 19 \beta_{9} + 9 \beta_{10} + 2 \beta_{14} - \beta_{16} + \beta_{17} - 10 \beta_{18} ) q^{67} + ( 341 \beta_{1} + 121 \beta_{2} + 35 \beta_{3} + 18 \beta_{6} - 21 \beta_{9} - 2 \beta_{10} - 13 \beta_{14} - 11 \beta_{16} - 6 \beta_{17} - 5 \beta_{18} ) q^{69} + ( 10026 + 20 \beta_{4} + 36 \beta_{5} - 26 \beta_{7} + 2 \beta_{8} + 20 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} - 6 \beta_{15} - 12 \beta_{19} ) q^{71} + ( -5238 + 21 \beta_{4} + 88 \beta_{5} + 3 \beta_{7} - \beta_{8} + 43 \beta_{11} + 2 \beta_{12} + 15 \beta_{15} + 5 \beta_{19} ) q^{73} + 625 \beta_{1} q^{75} + ( -780 \beta_{1} - 6 \beta_{2} + 32 \beta_{3} - 14 \beta_{6} - \beta_{9} - 5 \beta_{10} - 23 \beta_{14} - 3 \beta_{16} + 22 \beta_{17} - 3 \beta_{18} ) q^{77} + ( -14099 + 25 \beta_{4} - 10 \beta_{5} - 15 \beta_{7} - 3 \beta_{8} - 8 \beta_{12} - 10 \beta_{13} - 6 \beta_{15} - 11 \beta_{19} ) q^{79} + ( 3269 + 127 \beta_{4} - 87 \beta_{5} - 8 \beta_{7} - 2 \beta_{8} - 68 \beta_{11} + 8 \beta_{12} + 10 \beta_{13} - 6 \beta_{15} - 14 \beta_{19} ) q^{81} + ( 569 \beta_{1} - 168 \beta_{2} - 19 \beta_{3} - 42 \beta_{6} - 5 \beta_{9} + 13 \beta_{10} + 33 \beta_{14} + 15 \beta_{16} - 23 \beta_{17} + 35 \beta_{18} ) q^{83} + ( -257 \beta_{1} - 24 \beta_{2} - 35 \beta_{3} - 30 \beta_{6} + 19 \beta_{9} + 11 \beta_{10} - \beta_{14} + 9 \beta_{16} - \beta_{17} + 6 \beta_{18} ) q^{85} + ( 16619 - \beta_{4} - 28 \beta_{5} + 28 \beta_{7} - 2 \beta_{8} - 15 \beta_{11} + 26 \beta_{12} + 7 \beta_{13} + 9 \beta_{15} - 11 \beta_{19} ) q^{87} + ( -191 - 11 \beta_{4} - 139 \beta_{5} - 8 \beta_{7} - 2 \beta_{8} + 86 \beta_{11} + 21 \beta_{12} - 11 \beta_{13} - 13 \beta_{15} + 22 \beta_{19} ) q^{89} + ( -1348 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} + 13 \beta_{6} - 18 \beta_{9} - 18 \beta_{10} - 23 \beta_{14} - 30 \beta_{16} + 6 \beta_{17} - 8 \beta_{18} ) q^{91} + ( 443 \beta_{1} - 609 \beta_{2} - 54 \beta_{3} + 14 \beta_{6} + 56 \beta_{9} + 23 \beta_{10} + 81 \beta_{14} + 15 \beta_{16} - 14 \beta_{17} + 29 \beta_{18} ) q^{93} + ( -7220 - 45 \beta_{4} + 70 \beta_{5} + 20 \beta_{7} - 21 \beta_{11} + 10 \beta_{12} - 10 \beta_{13} + 5 \beta_{15} + 5 \beta_{19} ) q^{95} + ( 7346 - 195 \beta_{4} + 26 \beta_{5} + \beta_{7} - 3 \beta_{8} - 127 \beta_{11} - 30 \beta_{12} - 4 \beta_{13} - 23 \beta_{15} - \beta_{19} ) q^{97} + ( -101 \beta_{1} - 392 \beta_{2} - 121 \beta_{3} - 3 \beta_{6} + 160 \beta_{9} - 32 \beta_{10} + \beta_{14} + 28 \beta_{16} + 14 \beta_{17} - 14 \beta_{18} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 196q^{7} - 1620q^{9} + O(q^{10}) \) \( 20q + 196q^{7} - 1620q^{9} - 900q^{15} + 4676q^{23} - 12500q^{25} - 7160q^{31} + 5672q^{33} + 44904q^{39} + 11608q^{41} - 44180q^{47} + 18756q^{49} + 24200q^{55} + 5032q^{57} - 240620q^{63} + 200312q^{71} - 105136q^{73} - 282080q^{79} + 65172q^{81} + 332592q^{87} - 3160q^{89} - 144400q^{95} + 147376q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} - 109104 x^{12} - 96128 x^{11} + 3580672 x^{10} - 1538048 x^{9} - 27930624 x^{8} + 79364096 x^{7} + 157024256 x^{6} - 926941184 x^{5} + 4244635648 x^{4} + 20937965568 x^{3} - 73014444032 x^{2} - 137438953472 x + 1099511627776\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-79348087 \nu^{19} + 164467438 \nu^{18} + 2750967783 \nu^{17} + 2347892670 \nu^{16} - 17200470427 \nu^{15} + 69605863660 \nu^{14} - 24303499972 \nu^{13} - 719687080656 \nu^{12} + 16148446793808 \nu^{11} + 66989955357824 \nu^{10} - 209376218715392 \nu^{9} - 308378042677248 \nu^{8} + 2858199953264640 \nu^{7} + 4761986870738944 \nu^{6} + 7314399222824960 \nu^{5} + 210707338900799488 \nu^{4} + 19270179452092416 \nu^{3} - 2314359262031118336 \nu^{2} + 3379581800966782976 \nu + 38936417104430104576\)\()/ 713485464769658880 \)
\(\beta_{2}\)\(=\)\((\)\(-6673115 \nu^{19} - 30164170 \nu^{18} + 153706635 \nu^{17} + 266710470 \nu^{16} - 3862360175 \nu^{15} - 2106440260 \nu^{14} - 4737874100 \nu^{13} - 236780073360 \nu^{12} + 50299499280 \nu^{11} + 4828965765760 \nu^{10} - 12397823115520 \nu^{9} - 82271702046720 \nu^{8} + 93706914631680 \nu^{7} + 50248724971520 \nu^{6} - 1926810069893120 \nu^{5} + 7838825908797440 \nu^{4} + 6061205571502080 \nu^{3} - 180832148914176000 \nu^{2} - 275660321136312320 \nu + 1550266383906897920\)\()/ 32930098373984256 \)
\(\beta_{3}\)\(=\)\((\)\(392347319 \nu^{19} - 3248676206 \nu^{18} - 37138350375 \nu^{17} - 56596678206 \nu^{16} + 116030366171 \nu^{15} - 215726096876 \nu^{14} - 7354614456892 \nu^{13} - 28326961252656 \nu^{12} - 89620996541520 \nu^{11} - 695657667918976 \nu^{10} + 64356184347904 \nu^{9} + 4877709075462144 \nu^{8} - 35110862377537536 \nu^{7} - 176315402435624960 \nu^{6} - 252028719960162304 \nu^{5} - 172884124007137280 \nu^{4} - 5395871712027869184 \nu^{3} + 4882972880129753088 \nu^{2} - 18049481855350079488 \nu - 464189588174098399232\)\()/ 713485464769658880 \)
\(\beta_{4}\)\(=\)\((\)\(-13756221 \nu^{19} - 7338982 \nu^{18} + 200878797 \nu^{17} - 336652854 \nu^{16} - 2577873801 \nu^{15} + 10357630916 \nu^{14} - 40516598892 \nu^{13} - 198756032368 \nu^{12} + 1182900850544 \nu^{11} + 3523610475392 \nu^{10} - 19032642164480 \nu^{9} - 17076135725056 \nu^{8} + 214125830696960 \nu^{7} - 344410878705664 \nu^{6} - 66398979883008 \nu^{5} + 14093127287570432 \nu^{4} - 25707255580065792 \nu^{3} - 64272954002767872 \nu^{2} + 140142841543262208 \nu + 1695627043779117056\)\()/ 13989911073914880 \)
\(\beta_{5}\)\(=\)\((\)\(16634889 \nu^{19} + 110715086 \nu^{18} - 463943001 \nu^{17} - 1145810466 \nu^{16} + 6592520229 \nu^{15} + 24985916492 \nu^{14} - 39165383748 \nu^{13} + 407299798448 \nu^{12} - 444635751856 \nu^{11} - 16455359245696 \nu^{10} + 31229684808448 \nu^{9} + 254075443275776 \nu^{8} - 337690320547840 \nu^{7} - 1102678310453248 \nu^{6} + 1472660100612096 \nu^{5} - 18613792178962432 \nu^{4} - 49157013485125632 \nu^{3} + 692899524384915456 \nu^{2} - 81055567702917120 \nu - 7971915667421003776\)\()/ 13989911073914880 \)
\(\beta_{6}\)\(=\)\((\)\(-1009053331 \nu^{19} + 5955093094 \nu^{18} + 92664246339 \nu^{17} + 23483121654 \nu^{16} - 677089321159 \nu^{15} + 3386184621532 \nu^{14} + 8948119875116 \nu^{13} + 28585511805168 \nu^{12} + 422451709184400 \nu^{11} + 1842365967659648 \nu^{10} - 6803910244993280 \nu^{9} - 7117010603464704 \nu^{8} + 163360662163329024 \nu^{7} + 266180090303021056 \nu^{6} + 667307502036844544 \nu^{5} + 4427672373038153728 \nu^{4} - 1299921691522105344 \nu^{3} - 48226044564681523200 \nu^{2} + 165267730063935143936 \nu + 1789430898294368763904\)\()/ 535114098577244160 \)
\(\beta_{7}\)\(=\)\((\)\(836343 \nu^{19} + 4825138 \nu^{18} + 26902617 \nu^{17} - 67997406 \nu^{16} - 250315557 \nu^{15} + 785814196 \nu^{14} + 4487363652 \nu^{13} + 26267975248 \nu^{12} + 131794625968 \nu^{11} - 71168402048 \nu^{10} - 2500521196288 \nu^{9} - 1606553298944 \nu^{8} + 47753989304320 \nu^{7} + 65189977194496 \nu^{6} + 147694426521600 \nu^{5} + 212935867629568 \nu^{4} - 4662885082988544 \nu^{3} - 2307979059658752 \nu^{2} + 67795045155078144 \nu + 326224344046895104\)\()/ 241205363343360 \)
\(\beta_{8}\)\(=\)\((\)\(3296223 \nu^{19} - 144154430 \nu^{18} - 510836175 \nu^{17} + 2513179506 \nu^{16} - 302758557 \nu^{15} - 38622839180 \nu^{14} + 23498844708 \nu^{13} - 598187911088 \nu^{12} - 4937404206032 \nu^{11} + 6016066387840 \nu^{10} + 56099885206784 \nu^{9} - 258574748911616 \nu^{8} - 831571548753920 \nu^{7} + 138450305351680 \nu^{6} - 10419102624251904 \nu^{5} - 7726509879984128 \nu^{4} + 131950309923618816 \nu^{3} - 207790633207726080 \nu^{2} - 3088787054461452288 \nu - 2698545235025199104\)\()/ 874369442119680 \)
\(\beta_{9}\)\(=\)\((\)\(-4110998567 \nu^{19} + 2877435278 \nu^{18} + 134445630231 \nu^{17} + 31119529182 \nu^{16} - 1125456230411 \nu^{15} + 2395691636972 \nu^{14} - 1327532567300 \nu^{13} - 65684958912720 \nu^{12} + 749430500797776 \nu^{11} + 2601937693755520 \nu^{10} - 10940955354203392 \nu^{9} - 16243278503245824 \nu^{8} + 132142915823357952 \nu^{7} + 69003910881148928 \nu^{6} + 910441646450802688 \nu^{5} + 7670849992021508096 \nu^{4} - 9197198500901683200 \nu^{3} - 96771021572960944128 \nu^{2} + 283901422599109869568 \nu + 1261662170434487975936\)\()/ 1070228197154488320 \)
\(\beta_{10}\)\(=\)\((\)\(260801237 \nu^{19} - 1824079658 \nu^{18} - 2872664613 \nu^{17} + 20832970854 \nu^{16} - 17140065151 \nu^{15} - 40099565828 \nu^{14} - 527426412916 \nu^{13} - 6963291085968 \nu^{12} - 19077433527792 \nu^{11} + 68364900248192 \nu^{10} + 239411450138368 \nu^{9} - 2610189305567232 \nu^{8} - 2069946286534656 \nu^{7} - 23652572186869760 \nu^{6} - 84020616738635776 \nu^{5} + 108333149439655936 \nu^{4} - 87124716917096448 \nu^{3} - 8007115074579726336 \nu^{2} - 20198180721396809728 \nu - 13320290281937960960\)\()/ 54883497289973760 \)
\(\beta_{11}\)\(=\)\((\)\(5795805 \nu^{19} + 4381350 \nu^{18} - 62274925 \nu^{17} + 58776310 \nu^{16} + 1200456105 \nu^{15} + 95469500 \nu^{14} + 19123443180 \nu^{13} + 95638449520 \nu^{12} - 344796491120 \nu^{11} - 1303549488000 \nu^{10} + 9868738819840 \nu^{9} + 21718028912640 \nu^{8} - 8979288780800 \nu^{7} + 149102573977600 \nu^{6} + 60096778076160 \nu^{5} - 4584323499950080 \nu^{4} + 10719961295093760 \nu^{3} + 94815187999129600 \nu^{2} - 7978245349703680 \nu - 620428641750220800\)\()/ 932660738260992 \)
\(\beta_{12}\)\(=\)\((\)\(-1017903 \nu^{19} + 5922366 \nu^{18} + 40945759 \nu^{17} + 18273230 \nu^{16} - 696329523 \nu^{15} + 1014965932 \nu^{14} + 12216570780 \nu^{13} - 10837650640 \nu^{12} + 233765807312 \nu^{11} + 1469673965184 \nu^{10} - 4388269305088 \nu^{9} - 10438841456640 \nu^{8} + 89504132157440 \nu^{7} + 232292172234752 \nu^{6} - 248550134120448 \nu^{5} + 3349064691220480 \nu^{4} + 5978665091334144 \nu^{3} - 59776185520881664 \nu^{2} + 93891988000079872 \nu + 1056826868398817280\)\()/ 160803575562240 \)
\(\beta_{13}\)\(=\)\((\)\(-6553647 \nu^{19} + 4110910 \nu^{18} + 234956895 \nu^{17} - 334260274 \nu^{16} - 2157450547 \nu^{15} + 5693424300 \nu^{14} - 2662055012 \nu^{13} - 75569044688 \nu^{12} + 1156498253008 \nu^{11} + 4108959401600 \nu^{10} - 19669523737856 \nu^{9} - 30778824841216 \nu^{8} + 234445426708480 \nu^{7} + 143375976693760 \nu^{6} + 1313387938840576 \nu^{5} + 11563371375624192 \nu^{4} - 4002535035633664 \nu^{3} - 178325683846512640 \nu^{2} + 36119523908124672 \nu + 2851010354920554496\)\()/ 777217281884160 \)
\(\beta_{14}\)\(=\)\((\)\(-11729669627 \nu^{19} - 7756672522 \nu^{18} + 162936990891 \nu^{17} - 194149219962 \nu^{16} - 2398382102543 \nu^{15} - 3278410510660 \nu^{14} - 28704426242612 \nu^{13} - 254793725413776 \nu^{12} + 774486071124240 \nu^{11} + 4078553528162944 \nu^{10} - 21442484581097728 \nu^{9} - 60625577667385344 \nu^{8} + 112534678343651328 \nu^{7} - 26970025576431616 \nu^{6} - 581936411104772096 \nu^{5} + 10340798851136356352 \nu^{4} + 6747486828795789312 \nu^{3} - 293300664712524988416 \nu^{2} + 80374664847897395200 \nu + 2488771395149512048640\)\()/ 1070228197154488320 \)
\(\beta_{15}\)\(=\)\((\)\(57066063 \nu^{19} + 274798978 \nu^{18} - 936303743 \nu^{17} + 2504816114 \nu^{16} + 13924163283 \nu^{15} - 23482753324 \nu^{14} + 192206479332 \nu^{13} + 2782240740048 \nu^{12} + 1439053804848 \nu^{11} - 30347313231488 \nu^{10} + 88322490633472 \nu^{9} + 496240116422656 \nu^{8} - 152588912087040 \nu^{7} + 5925686790782976 \nu^{6} + 17277039999713280 \nu^{5} - 54335566891712512 \nu^{4} - 153856678847053824 \nu^{3} + 1497369574205554688 \nu^{2} + 4219199795111133184 \nu - 5100257240045060096\)\()/ 4663303691304960 \)
\(\beta_{16}\)\(=\)\((\)\(445476863 \nu^{19} + 755206978 \nu^{18} - 2379736431 \nu^{17} + 690187890 \nu^{16} + 41497627523 \nu^{15} + 424482480052 \nu^{14} + 2364490799780 \nu^{13} + 8491155454800 \nu^{12} - 2213726830032 \nu^{11} - 135462442775680 \nu^{10} + 443775654153472 \nu^{9} + 3113792848312320 \nu^{8} + 7601811276902400 \nu^{7} + 8147471708520448 \nu^{6} + 7364567046029312 \nu^{5} - 78951472682762240 \nu^{4} + 130657928091795456 \nu^{3} + 9577769516843139072 \nu^{2} + 24592412889225101312 \nu - 55918171747851960320\)\()/ 31477299916308480 \)
\(\beta_{17}\)\(=\)\((\)\(-279607999 \nu^{19} + 207980926 \nu^{18} + 4651875759 \nu^{17} - 5763844146 \nu^{16} - 69380146627 \nu^{15} - 15389892212 \nu^{14} - 180631027876 \nu^{13} - 5214217274448 \nu^{12} + 24807733624272 \nu^{11} + 58583395989632 \nu^{10} - 593749860183296 \nu^{9} - 880987800532992 \nu^{8} + 2653284081782784 \nu^{7} + 1255725749960704 \nu^{6} - 25940664486461440 \nu^{5} + 165537541947129856 \nu^{4} - 637830318060994560 \nu^{3} - 6394634195309690880 \nu^{2} + 7376531761657806848 \nu + 21477900337870274560\)\()/ 18294499096657920 \)
\(\beta_{18}\)\(=\)\((\)\(-34415252287 \nu^{19} + 44118579838 \nu^{18} + 614301056559 \nu^{17} - 769774935858 \nu^{16} - 4977821113411 \nu^{15} - 2337983947892 \nu^{14} - 48612162875044 \nu^{13} - 299854001953872 \nu^{12} + 2489242855970256 \nu^{11} + 9428986422343808 \nu^{10} - 59405241520221440 \nu^{9} - 80251333231441920 \nu^{8} + 266111322945933312 \nu^{7} - 85505550715781120 \nu^{6} + 1030837898012524544 \nu^{5} + 13514949335179067392 \nu^{4} - 21716798153781411840 \nu^{3} - 471425196044209618944 \nu^{2} - 274160655454336712704 \nu + 4667308348567283826688\)\()/ 2140456394308976640 \)
\(\beta_{19}\)\(=\)\((\)\(240970995 \nu^{19} + 955006394 \nu^{18} - 2146959459 \nu^{17} - 1890330006 \nu^{16} + 60572738535 \nu^{15} + 136049634308 \nu^{14} + 605040164052 \nu^{13} + 7864865425808 \nu^{12} + 7262658201200 \nu^{11} - 95866524978304 \nu^{10} + 297809594185984 \nu^{9} + 2218657466710016 \nu^{8} + 1657499988684800 \nu^{7} + 6687192417763328 \nu^{6} + 58820173022625792 \nu^{5} - 45042636666437632 \nu^{4} - 419669485478215680 \nu^{3} + 4829801159772340224 \nu^{2} + 13555780229221318656 \nu - 22273251727592390656\)\()/ 13989911073914880 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(4 \beta_{19} - 14 \beta_{18} - \beta_{17} - 6 \beta_{16} - 9 \beta_{15} - \beta_{14} - 21 \beta_{13} + 5 \beta_{12} - 28 \beta_{11} - 4 \beta_{10} + 14 \beta_{9} - 4 \beta_{8} + 4 \beta_{7} - 10 \beta_{6} - 31 \beta_{5} - 15 \beta_{4} - 45 \beta_{3} + 2 \beta_{2} - 52 \beta_{1} + 1269\)\()/12800\)
\(\nu^{2}\)\(=\)\((\)\(-44 \beta_{19} + 14 \beta_{18} - 39 \beta_{17} + 6 \beta_{16} + 19 \beta_{15} - 19 \beta_{14} - 9 \beta_{13} + 5 \beta_{12} + 84 \beta_{11} - 36 \beta_{10} + 106 \beta_{9} + 4 \beta_{8} + 56 \beta_{7} - 50 \beta_{6} + \beta_{5} + 445 \beta_{4} + 125 \beta_{3} - 378 \beta_{2} + 1252 \beta_{1} + 24401\)\()/12800\)
\(\nu^{3}\)\(=\)\((\)\(-84 \beta_{19} + 18 \beta_{18} - 108 \beta_{17} + 112 \beta_{16} + 39 \beta_{15} + 192 \beta_{14} + 41 \beta_{13} - 5 \beta_{12} - 126 \beta_{11} + 28 \beta_{10} - 68 \beta_{9} - 16 \beta_{8} - 34 \beta_{7} - 110 \beta_{6} + 251 \beta_{5} - 985 \beta_{4} - 220 \beta_{3} + 212 \beta_{2} + 2714 \beta_{1} - 39819\)\()/6400\)
\(\nu^{4}\)\(=\)\((\)\(36 \beta_{19} - 10 \beta_{18} - \beta_{17} + 10 \beta_{16} - 17 \beta_{15} + 11 \beta_{14} + 7 \beta_{13} - 11 \beta_{12} - 88 \beta_{11} - 36 \beta_{10} - 34 \beta_{9} + 12 \beta_{8} + 28 \beta_{7} - 2 \beta_{6} - 71 \beta_{5} - 75 \beta_{4} + 43 \beta_{3} - 30 \beta_{2} + 1776 \beta_{1} - 19739\)\()/512\)
\(\nu^{5}\)\(=\)\((\)\(68 \beta_{19} - 654 \beta_{18} - 911 \beta_{17} + 134 \beta_{16} + 347 \beta_{15} + 1289 \beta_{14} + 2843 \beta_{13} - 515 \beta_{12} - 3920 \beta_{11} - 1044 \beta_{10} + 9754 \beta_{9} + 532 \beta_{8} - 2232 \beta_{7} + 3790 \beta_{6} + 13073 \beta_{5} - 35655 \beta_{4} - 195 \beta_{3} - 218 \beta_{2} - 289672 \beta_{1} - 1730847\)\()/12800\)
\(\nu^{6}\)\(=\)\((\)\(-3196 \beta_{19} - 10606 \beta_{18} + 7856 \beta_{17} - 5024 \beta_{16} + 1491 \beta_{15} - 1324 \beta_{14} + 4689 \beta_{13} - 2585 \beta_{12} + 18014 \beta_{11} - 7356 \beta_{10} - 8124 \beta_{9} + 296 \beta_{8} - 2066 \beta_{7} + 9850 \beta_{6} + 14639 \beta_{5} - 2185 \beta_{4} + 6400 \beta_{3} + 43172 \beta_{2} + 285842 \beta_{1} - 9927991\)\()/6400\)
\(\nu^{7}\)\(=\)\((\)\(16324 \beta_{19} + 49938 \beta_{18} - 14313 \beta_{17} + 19162 \beta_{16} + 14471 \beta_{15} + 85487 \beta_{14} - 15301 \beta_{13} - 9795 \beta_{12} + 106052 \beta_{11} + 27388 \beta_{10} + 5262 \beta_{9} + 10476 \beta_{8} + 8124 \beta_{7} + 71150 \beta_{6} + 16089 \beta_{5} + 514385 \beta_{4} + 84475 \beta_{3} - 125070 \beta_{2} - 2814756 \beta_{1} - 78838411\)\()/12800\)
\(\nu^{8}\)\(=\)\((\)\(2676 \beta_{19} + 4086 \beta_{18} + 3473 \beta_{17} - 10 \beta_{16} + 1195 \beta_{15} - 1331 \beta_{14} + 3991 \beta_{13} - 1923 \beta_{12} + 21532 \beta_{11} - 1412 \beta_{10} - 13574 \beta_{9} - 1532 \beta_{8} - 14040 \beta_{7} + 6094 \beta_{6} - 7767 \beta_{5} - 54467 \beta_{4} + 28837 \beta_{3} + 410198 \beta_{2} + 227036 \beta_{1} + 6757289\)\()/512\)
\(\nu^{9}\)\(=\)\((\)\(525340 \beta_{19} - 91758 \beta_{18} + 136748 \beta_{17} - 210672 \beta_{16} - 145665 \beta_{15} + 233448 \beta_{14} - 86535 \beta_{13} + 41875 \beta_{12} + 372282 \beta_{11} + 194732 \beta_{10} + 98508 \beta_{9} + 16160 \beta_{8} - 71810 \beta_{7} - 347390 \beta_{6} - 120285 \beta_{5} + 4778375 \beta_{4} - 1566980 \beta_{3} + 8138228 \beta_{2} - 22020534 \beta_{1} + 480581725\)\()/6400\)
\(\nu^{10}\)\(=\)\((\)\(325380 \beta_{19} + 731134 \beta_{18} - 1210009 \beta_{17} - 3982214 \beta_{16} - 936425 \beta_{15} - 6039989 \beta_{14} - 2324665 \beta_{13} + 2845805 \beta_{12} + 6414912 \beta_{11} + 220284 \beta_{10} + 3760686 \beta_{9} - 1129940 \beta_{8} - 1680900 \beta_{7} - 2164850 \beta_{6} - 9232575 \beta_{5} + 15202165 \beta_{4} + 5661475 \beta_{3} + 63180562 \beta_{2} - 25585688 \beta_{1} - 2969570195\)\()/12800\)
\(\nu^{11}\)\(=\)\((\)\(-13704092 \beta_{19} + 5590458 \beta_{18} - 4960383 \beta_{17} + 5939142 \beta_{16} + 13609707 \beta_{15} + 3599617 \beta_{14} + 13910883 \beta_{13} + 3487885 \beta_{12} + 30064360 \beta_{11} + 14815308 \beta_{10} + 6984442 \beta_{9} - 1260108 \beta_{8} + 889608 \beta_{7} - 30912850 \beta_{6} - 2883887 \beta_{5} + 146464945 \beta_{4} + 764125 \beta_{3} - 291693850 \beta_{2} + 186641504 \beta_{1} + 13219713793\)\()/12800\)
\(\nu^{12}\)\(=\)\((\)\(526228 \beta_{19} + 377730 \beta_{18} - 609080 \beta_{17} + 191424 \beta_{16} - 227613 \beta_{15} - 677156 \beta_{14} - 400887 \beta_{13} + 325399 \beta_{12} - 4508218 \beta_{11} + 575860 \beta_{10} - 3495116 \beta_{9} - 200168 \beta_{8} + 293886 \beta_{7} - 897510 \beta_{6} - 5893585 \beta_{5} + 1560751 \beta_{4} - 2596520 \beta_{3} - 8567708 \beta_{2} + 25492274 \beta_{1} - 1250502055\)\()/256\)
\(\nu^{13}\)\(=\)\((\)\(-14642492 \beta_{19} + 103278602 \beta_{18} + 45947223 \beta_{17} + 146964698 \beta_{16} - 13739193 \beta_{15} - 274655577 \beta_{14} + 167747283 \beta_{13} + 59506685 \beta_{12} - 363730660 \beta_{11} - 19081348 \beta_{10} + 268349198 \beta_{9} + 54022892 \beta_{8} + 15173308 \beta_{7} - 142812050 \beta_{6} - 602015287 \beta_{5} - 286535655 \beta_{4} + 132428075 \beta_{3} - 8925510030 \beta_{2} - 8029473324 \beta_{1} + 272916930693\)\()/12800\)
\(\nu^{14}\)\(=\)\((\)\(-610849420 \beta_{19} - 1191174514 \beta_{18} + 436423289 \beta_{17} - 95003706 \beta_{16} - 268281725 \beta_{15} - 994905731 \beta_{14} + 79948935 \beta_{13} + 103955205 \beta_{12} - 2654362828 \beta_{11} + 149850236 \beta_{10} - 341290806 \beta_{9} - 108402940 \beta_{8} - 428176600 \beta_{7} + 779741950 \beta_{6} + 3793620225 \beta_{5} - 4026719635 \beta_{4} - 3556998275 \beta_{3} - 21797731162 \beta_{2} + 7334948 \beta_{1} - 341144190495\)\()/12800\)
\(\nu^{15}\)\(=\)\((\)\(-871984500 \beta_{19} - 160118542 \beta_{18} + 880534852 \beta_{17} - 86269328 \beta_{16} - 506109225 \beta_{15} - 680664048 \beta_{14} - 869833975 \beta_{13} - 620805525 \beta_{12} + 1940428082 \beta_{11} - 653569732 \beta_{10} + 179384092 \beta_{9} + 437480400 \beta_{8} + 1615280350 \beta_{7} + 1956387090 \beta_{6} - 4627899125 \beta_{5} + 9640071975 \beta_{4} + 2470103380 \beta_{3} - 135376526828 \beta_{2} + 99281887834 \beta_{1} - 1625292214715\)\()/6400\)
\(\nu^{16}\)\(=\)\((\)\(-408459356 \beta_{19} + 447320870 \beta_{18} - 120078593 \beta_{17} + 597058826 \beta_{16} + 373970095 \beta_{15} + 404508155 \beta_{14} - 145902025 \beta_{13} - 287005067 \beta_{12} - 513172456 \beta_{11} + 377274204 \beta_{10} + 399285342 \beta_{9} + 62833164 \beta_{8} - 308012196 \beta_{7} + 640417470 \beta_{6} + 2260962329 \beta_{5} - 5024644539 \beta_{4} + 90840331 \beta_{3} + 2217999970 \beta_{2} + 13561143584 \beta_{1} + 326130296645\)\()/512\)
\(\nu^{17}\)\(=\)\((\)\(42668508804 \beta_{19} - 14442191358 \beta_{18} + 46349039633 \beta_{17} + 2523089158 \beta_{16} - 39595954309 \beta_{15} + 18573143033 \beta_{14} - 20870356821 \beta_{13} - 55490564195 \beta_{12} + 38806841632 \beta_{11} - 36430765908 \beta_{10} - 71473967142 \beta_{9} + 1357837396 \beta_{8} + 21933976904 \beta_{7} + 18461223950 \beta_{6} - 36070946031 \beta_{5} - 412462979415 \beta_{4} - 143906710275 \beta_{3} + 1840354413990 \beta_{2} + 3895031802696 \beta_{1} + 54880897505569\)\()/12800\)
\(\nu^{18}\)\(=\)\((\)\(79647000164 \beta_{19} + 72314538066 \beta_{18} - 46315267616 \beta_{17} - 50923610336 \beta_{16} - 46613619069 \beta_{15} + 68569046564 \beta_{14} - 145785265551 \beta_{13} + 42593445015 \beta_{12} - 35420251826 \beta_{11} - 4963779484 \beta_{10} + 314785960164 \beta_{9} + 15303372936 \beta_{8} + 81780472494 \beta_{7} - 3031464550 \beta_{6} + 734392870399 \beta_{5} - 459204576585 \beta_{4} - 162479678800 \beta_{3} + 2226298022628 \beta_{2} - 4383724445262 \beta_{1} + 31192842076569\)\()/6400\)
\(\nu^{19}\)\(=\)\((\)\(-417829414972 \beta_{19} - 400451474046 \beta_{18} - 181387347689 \beta_{17} - 512963191334 \beta_{16} + 119021860487 \beta_{15} - 54094601889 \beta_{14} + 205611761003 \beta_{13} - 62903746115 \beta_{12} + 3357788221300 \beta_{11} - 23728061956 \beta_{10} + 971350946446 \beta_{9} - 236613835028 \beta_{8} + 381057629628 \beta_{7} - 549069993490 \beta_{6} + 2266265144633 \beta_{5} - 9631575098655 \beta_{4} + 2731604216795 \beta_{3} + 41915195352178 \beta_{2} + 56906488516172 \beta_{1} + 255280619285013\)\()/12800\)

Character values

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

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
3.46430 + 1.99965i
−3.90102 0.884346i
0.236693 3.99299i
2.93366 + 2.71913i
3.18502 2.41984i
−3.80026 1.24819i
3.72553 1.45618i
−2.80358 + 2.85306i
0.593959 + 3.95566i
−2.63430 + 3.01006i
−2.63430 3.01006i
0.593959 3.95566i
−2.80358 2.85306i
3.72553 + 1.45618i
−3.80026 + 1.24819i
3.18502 + 2.41984i
2.93366 2.71913i
0.236693 + 3.99299i
−3.90102 + 0.884346i
3.46430 1.99965i
0 29.2080i 0 25.0000i 0 168.173 0 −610.110 0
81.2 0 25.4343i 0 25.0000i 0 56.4938 0 −403.904 0
81.3 0 25.0521i 0 25.0000i 0 −103.624 0 −384.607 0
81.4 0 18.7876i 0 25.0000i 0 107.536 0 −109.975 0
81.5 0 17.3148i 0 25.0000i 0 9.19080 0 −56.8021 0
81.6 0 11.5927i 0 25.0000i 0 231.529 0 108.609 0
81.7 0 10.8240i 0 25.0000i 0 −163.706 0 125.841 0
81.8 0 10.7455i 0 25.0000i 0 −198.733 0 127.535 0
81.9 0 6.93089i 0 25.0000i 0 −47.1406 0 194.963 0
81.10 0 6.67450i 0 25.0000i 0 38.2812 0 198.451 0
81.11 0 6.67450i 0 25.0000i 0 38.2812 0 198.451 0
81.12 0 6.93089i 0 25.0000i 0 −47.1406 0 194.963 0
81.13 0 10.7455i 0 25.0000i 0 −198.733 0 127.535 0
81.14 0 10.8240i 0 25.0000i 0 −163.706 0 125.841 0
81.15 0 11.5927i 0 25.0000i 0 231.529 0 108.609 0
81.16 0 17.3148i 0 25.0000i 0 9.19080 0 −56.8021 0
81.17 0 18.7876i 0 25.0000i 0 107.536 0 −109.975 0
81.18 0 25.0521i 0 25.0000i 0 −103.624 0 −384.607 0
81.19 0 25.4343i 0 25.0000i 0 56.4938 0 −403.904 0
81.20 0 29.2080i 0 25.0000i 0 168.173 0 −610.110 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.6.d.a 20
4.b odd 2 1 40.6.d.a 20
5.b even 2 1 800.6.d.c 20
5.c odd 4 1 800.6.f.b 20
5.c odd 4 1 800.6.f.c 20
8.b even 2 1 inner 160.6.d.a 20
8.d odd 2 1 40.6.d.a 20
12.b even 2 1 360.6.k.b 20
20.d odd 2 1 200.6.d.b 20
20.e even 4 1 200.6.f.b 20
20.e even 4 1 200.6.f.c 20
24.f even 2 1 360.6.k.b 20
40.e odd 2 1 200.6.d.b 20
40.f even 2 1 800.6.d.c 20
40.i odd 4 1 800.6.f.b 20
40.i odd 4 1 800.6.f.c 20
40.k even 4 1 200.6.f.b 20
40.k even 4 1 200.6.f.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.d.a 20 4.b odd 2 1
40.6.d.a 20 8.d odd 2 1
160.6.d.a 20 1.a even 1 1 trivial
160.6.d.a 20 8.b even 2 1 inner
200.6.d.b 20 20.d odd 2 1
200.6.d.b 20 40.e odd 2 1
200.6.f.b 20 20.e even 4 1
200.6.f.b 20 40.k even 4 1
200.6.f.c 20 20.e even 4 1
200.6.f.c 20 40.k even 4 1
360.6.k.b 20 12.b even 2 1
360.6.k.b 20 24.f even 2 1
800.6.d.c 20 5.b even 2 1
800.6.d.c 20 40.f even 2 1
800.6.f.b 20 5.c odd 4 1
800.6.f.b 20 40.i odd 4 1
800.6.f.c 20 5.c odd 4 1
800.6.f.c 20 40.i odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(160, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 1620 T^{2} + 1394322 T^{4} - 834927892 T^{6} + 392823148221 T^{8} - 155295909553872 T^{10} + 53814329151823576 T^{12} - 16772396183204761872 T^{14} + \)\(47\!\cdots\!30\)\( T^{16} - \)\(12\!\cdots\!00\)\( T^{18} + \)\(31\!\cdots\!00\)\( T^{20} - \)\(75\!\cdots\!00\)\( T^{22} + \)\(16\!\cdots\!30\)\( T^{24} - \)\(34\!\cdots\!28\)\( T^{26} + \)\(65\!\cdots\!76\)\( T^{28} - \)\(11\!\cdots\!28\)\( T^{30} + \)\(16\!\cdots\!21\)\( T^{32} - \)\(20\!\cdots\!08\)\( T^{34} + \)\(20\!\cdots\!22\)\( T^{36} - \)\(14\!\cdots\!80\)\( T^{38} + \)\(51\!\cdots\!01\)\( T^{40} \)
$5$ \( ( 1 + 625 T^{2} )^{10} \)
$7$ \( ( 1 - 98 T + 84148 T^{2} - 8017214 T^{3} + 3556570901 T^{4} - 345044190776 T^{5} + 103920201897616 T^{6} - 10159390994080936 T^{7} + 2363994840482709802 T^{8} - \)\(22\!\cdots\!76\)\( T^{9} + \)\(43\!\cdots\!72\)\( T^{10} - \)\(37\!\cdots\!32\)\( T^{11} + \)\(66\!\cdots\!98\)\( T^{12} - \)\(48\!\cdots\!48\)\( T^{13} + \)\(82\!\cdots\!16\)\( T^{14} - \)\(46\!\cdots\!32\)\( T^{15} + \)\(80\!\cdots\!49\)\( T^{16} - \)\(30\!\cdots\!02\)\( T^{17} + \)\(53\!\cdots\!48\)\( T^{18} - \)\(10\!\cdots\!86\)\( T^{19} + \)\(17\!\cdots\!49\)\( T^{20} )^{2} \)
$11$ \( 1 - 1510004 T^{2} + 1155249727726 T^{4} - 592766540112799924 T^{6} + \)\(22\!\cdots\!17\)\( T^{8} - \)\(70\!\cdots\!04\)\( T^{10} + \)\(18\!\cdots\!36\)\( T^{12} - \)\(40\!\cdots\!64\)\( T^{14} + \)\(79\!\cdots\!82\)\( T^{16} - \)\(14\!\cdots\!04\)\( T^{18} + \)\(23\!\cdots\!76\)\( T^{20} - \)\(36\!\cdots\!04\)\( T^{22} + \)\(53\!\cdots\!82\)\( T^{24} - \)\(70\!\cdots\!64\)\( T^{26} + \)\(82\!\cdots\!36\)\( T^{28} - \)\(82\!\cdots\!04\)\( T^{30} + \)\(69\!\cdots\!17\)\( T^{32} - \)\(46\!\cdots\!24\)\( T^{34} + \)\(23\!\cdots\!26\)\( T^{36} - \)\(80\!\cdots\!04\)\( T^{38} + \)\(13\!\cdots\!01\)\( T^{40} \)
$13$ \( 1 - 3520332 T^{2} + 6227853839054 T^{4} - 7441447313525468748 T^{6} + \)\(67\!\cdots\!57\)\( T^{8} - \)\(50\!\cdots\!12\)\( T^{10} + \)\(32\!\cdots\!64\)\( T^{12} - \)\(17\!\cdots\!28\)\( T^{14} + \)\(87\!\cdots\!42\)\( T^{16} - \)\(38\!\cdots\!32\)\( T^{18} + \)\(14\!\cdots\!64\)\( T^{20} - \)\(52\!\cdots\!68\)\( T^{22} + \)\(16\!\cdots\!42\)\( T^{24} - \)\(46\!\cdots\!72\)\( T^{26} + \)\(11\!\cdots\!64\)\( T^{28} - \)\(25\!\cdots\!88\)\( T^{30} + \)\(46\!\cdots\!57\)\( T^{32} - \)\(70\!\cdots\!52\)\( T^{34} + \)\(81\!\cdots\!54\)\( T^{36} - \)\(63\!\cdots\!68\)\( T^{38} + \)\(24\!\cdots\!01\)\( T^{40} \)
$17$ \( ( 1 + 5447662 T^{2} + 1859072000 T^{3} + 15317572824845 T^{4} + 7413542230528000 T^{5} + 32518332783929091752 T^{6} + \)\(14\!\cdots\!00\)\( T^{7} + \)\(56\!\cdots\!10\)\( T^{8} + \)\(21\!\cdots\!00\)\( T^{9} + \)\(84\!\cdots\!72\)\( T^{10} + \)\(31\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!90\)\( T^{12} + \)\(41\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!52\)\( T^{14} + \)\(42\!\cdots\!00\)\( T^{15} + \)\(12\!\cdots\!05\)\( T^{16} + \)\(21\!\cdots\!00\)\( T^{17} + \)\(89\!\cdots\!62\)\( T^{18} + \)\(33\!\cdots\!49\)\( T^{20} )^{2} \)
$19$ \( 1 - 23638692 T^{2} + 296336418074734 T^{4} - \)\(25\!\cdots\!32\)\( T^{6} + \)\(17\!\cdots\!97\)\( T^{8} - \)\(93\!\cdots\!80\)\( T^{10} + \)\(42\!\cdots\!52\)\( T^{12} - \)\(16\!\cdots\!48\)\( T^{14} + \)\(56\!\cdots\!66\)\( T^{16} - \)\(16\!\cdots\!12\)\( T^{18} + \)\(44\!\cdots\!28\)\( T^{20} - \)\(10\!\cdots\!12\)\( T^{22} + \)\(21\!\cdots\!66\)\( T^{24} - \)\(38\!\cdots\!48\)\( T^{26} + \)\(60\!\cdots\!52\)\( T^{28} - \)\(80\!\cdots\!80\)\( T^{30} + \)\(91\!\cdots\!97\)\( T^{32} - \)\(83\!\cdots\!32\)\( T^{34} + \)\(59\!\cdots\!34\)\( T^{36} - \)\(28\!\cdots\!92\)\( T^{38} + \)\(75\!\cdots\!01\)\( T^{40} \)
$23$ \( ( 1 - 2338 T + 35997660 T^{2} - 45007042654 T^{3} + 520972471777845 T^{4} - 50426407997609208 T^{5} + \)\(39\!\cdots\!60\)\( T^{6} + \)\(66\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!10\)\( T^{8} + \)\(91\!\cdots\!52\)\( T^{9} + \)\(76\!\cdots\!60\)\( T^{10} + \)\(58\!\cdots\!36\)\( T^{11} + \)\(74\!\cdots\!90\)\( T^{12} + \)\(17\!\cdots\!72\)\( T^{13} + \)\(68\!\cdots\!60\)\( T^{14} - \)\(55\!\cdots\!44\)\( T^{15} + \)\(37\!\cdots\!05\)\( T^{16} - \)\(20\!\cdots\!78\)\( T^{17} + \)\(10\!\cdots\!60\)\( T^{18} - \)\(44\!\cdots\!34\)\( T^{19} + \)\(12\!\cdots\!49\)\( T^{20} )^{2} \)
$29$ \( 1 - 215092900 T^{2} + 23243889276296494 T^{4} - \)\(16\!\cdots\!00\)\( T^{6} + \)\(90\!\cdots\!13\)\( T^{8} - \)\(38\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!92\)\( T^{12} - \)\(42\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!86\)\( T^{16} - \)\(27\!\cdots\!00\)\( T^{18} + \)\(58\!\cdots\!28\)\( T^{20} - \)\(11\!\cdots\!00\)\( T^{22} + \)\(20\!\cdots\!86\)\( T^{24} - \)\(31\!\cdots\!00\)\( T^{26} + \)\(43\!\cdots\!92\)\( T^{28} - \)\(51\!\cdots\!00\)\( T^{30} + \)\(50\!\cdots\!13\)\( T^{32} - \)\(39\!\cdots\!00\)\( T^{34} + \)\(22\!\cdots\!94\)\( T^{36} - \)\(88\!\cdots\!00\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( ( 1 + 3580 T + 132421614 T^{2} + 484936307876 T^{3} + 8983138546835629 T^{4} + 33755686429634218608 T^{5} + \)\(43\!\cdots\!88\)\( T^{6} + \)\(16\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!38\)\( T^{8} + \)\(60\!\cdots\!32\)\( T^{9} + \)\(54\!\cdots\!44\)\( T^{10} + \)\(17\!\cdots\!32\)\( T^{11} + \)\(13\!\cdots\!38\)\( T^{12} + \)\(38\!\cdots\!96\)\( T^{13} + \)\(29\!\cdots\!88\)\( T^{14} + \)\(64\!\cdots\!08\)\( T^{15} + \)\(49\!\cdots\!29\)\( T^{16} + \)\(76\!\cdots\!76\)\( T^{17} + \)\(59\!\cdots\!14\)\( T^{18} + \)\(46\!\cdots\!80\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$37$ \( 1 - 565372668 T^{2} + 171401952644913934 T^{4} - \)\(36\!\cdots\!00\)\( T^{6} + \)\(59\!\cdots\!09\)\( T^{8} - \)\(80\!\cdots\!44\)\( T^{10} + \)\(93\!\cdots\!16\)\( T^{12} - \)\(93\!\cdots\!40\)\( T^{14} + \)\(83\!\cdots\!86\)\( T^{16} - \)\(67\!\cdots\!48\)\( T^{18} + \)\(48\!\cdots\!08\)\( T^{20} - \)\(32\!\cdots\!52\)\( T^{22} + \)\(19\!\cdots\!86\)\( T^{24} - \)\(10\!\cdots\!60\)\( T^{26} + \)\(49\!\cdots\!16\)\( T^{28} - \)\(20\!\cdots\!56\)\( T^{30} + \)\(73\!\cdots\!09\)\( T^{32} - \)\(21\!\cdots\!00\)\( T^{34} + \)\(48\!\cdots\!34\)\( T^{36} - \)\(77\!\cdots\!32\)\( T^{38} + \)\(66\!\cdots\!01\)\( T^{40} \)
$41$ \( ( 1 - 5804 T + 580737234 T^{2} - 4176614475628 T^{3} + 181136735899530493 T^{4} - \)\(13\!\cdots\!48\)\( T^{5} + \)\(39\!\cdots\!40\)\( T^{6} - \)\(28\!\cdots\!68\)\( T^{7} + \)\(63\!\cdots\!42\)\( T^{8} - \)\(43\!\cdots\!24\)\( T^{9} + \)\(82\!\cdots\!24\)\( T^{10} - \)\(49\!\cdots\!24\)\( T^{11} + \)\(85\!\cdots\!42\)\( T^{12} - \)\(44\!\cdots\!68\)\( T^{13} + \)\(70\!\cdots\!40\)\( T^{14} - \)\(28\!\cdots\!48\)\( T^{15} + \)\(43\!\cdots\!93\)\( T^{16} - \)\(11\!\cdots\!28\)\( T^{17} + \)\(18\!\cdots\!34\)\( T^{18} - \)\(21\!\cdots\!04\)\( T^{19} + \)\(43\!\cdots\!01\)\( T^{20} )^{2} \)
$43$ \( 1 - 1208126740 T^{2} + 787740581508660018 T^{4} - \)\(36\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!73\)\( T^{8} - \)\(37\!\cdots\!80\)\( T^{10} + \)\(95\!\cdots\!40\)\( T^{12} - \)\(20\!\cdots\!56\)\( T^{14} + \)\(40\!\cdots\!70\)\( T^{16} - \)\(70\!\cdots\!00\)\( T^{18} + \)\(10\!\cdots\!96\)\( T^{20} - \)\(15\!\cdots\!00\)\( T^{22} + \)\(18\!\cdots\!70\)\( T^{24} - \)\(21\!\cdots\!44\)\( T^{26} + \)\(20\!\cdots\!40\)\( T^{28} - \)\(17\!\cdots\!20\)\( T^{30} + \)\(13\!\cdots\!73\)\( T^{32} - \)\(79\!\cdots\!04\)\( T^{34} + \)\(37\!\cdots\!18\)\( T^{36} - \)\(12\!\cdots\!60\)\( T^{38} + \)\(22\!\cdots\!01\)\( T^{40} \)
$47$ \( ( 1 + 22090 T + 1548510076 T^{2} + 25779450599270 T^{3} + 1065218725316011845 T^{4} + \)\(14\!\cdots\!20\)\( T^{5} + \)\(45\!\cdots\!96\)\( T^{6} + \)\(51\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!10\)\( T^{8} + \)\(14\!\cdots\!00\)\( T^{9} + \)\(36\!\cdots\!56\)\( T^{10} + \)\(32\!\cdots\!00\)\( T^{11} + \)\(76\!\cdots\!90\)\( T^{12} + \)\(62\!\cdots\!80\)\( T^{13} + \)\(12\!\cdots\!96\)\( T^{14} + \)\(90\!\cdots\!40\)\( T^{15} + \)\(15\!\cdots\!05\)\( T^{16} + \)\(86\!\cdots\!10\)\( T^{17} + \)\(11\!\cdots\!76\)\( T^{18} + \)\(38\!\cdots\!30\)\( T^{19} + \)\(40\!\cdots\!49\)\( T^{20} )^{2} \)
$53$ \( 1 - 4003669356 T^{2} + 7954725909905649454 T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!77\)\( T^{8} - \)\(91\!\cdots\!76\)\( T^{10} + \)\(65\!\cdots\!16\)\( T^{12} - \)\(40\!\cdots\!80\)\( T^{14} + \)\(22\!\cdots\!54\)\( T^{16} - \)\(11\!\cdots\!56\)\( T^{18} + \)\(48\!\cdots\!96\)\( T^{20} - \)\(19\!\cdots\!44\)\( T^{22} + \)\(68\!\cdots\!54\)\( T^{24} - \)\(21\!\cdots\!20\)\( T^{26} + \)\(61\!\cdots\!16\)\( T^{28} - \)\(14\!\cdots\!24\)\( T^{30} + \)\(31\!\cdots\!77\)\( T^{32} - \)\(53\!\cdots\!40\)\( T^{34} + \)\(69\!\cdots\!54\)\( T^{36} - \)\(61\!\cdots\!44\)\( T^{38} + \)\(26\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 - 9996949828 T^{2} + 49650800837889885966 T^{4} - \)\(16\!\cdots\!20\)\( T^{6} + \)\(39\!\cdots\!17\)\( T^{8} - \)\(73\!\cdots\!28\)\( T^{10} + \)\(11\!\cdots\!44\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{14} + \)\(15\!\cdots\!54\)\( T^{16} - \)\(13\!\cdots\!88\)\( T^{18} + \)\(10\!\cdots\!24\)\( T^{20} - \)\(70\!\cdots\!88\)\( T^{22} + \)\(39\!\cdots\!54\)\( T^{24} - \)\(19\!\cdots\!80\)\( T^{26} + \)\(77\!\cdots\!44\)\( T^{28} - \)\(25\!\cdots\!28\)\( T^{30} + \)\(69\!\cdots\!17\)\( T^{32} - \)\(14\!\cdots\!20\)\( T^{34} + \)\(23\!\cdots\!66\)\( T^{36} - \)\(23\!\cdots\!28\)\( T^{38} + \)\(12\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 - 7152852348 T^{2} + 26523669582205677166 T^{4} - \)\(67\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!17\)\( T^{8} - \)\(22\!\cdots\!48\)\( T^{10} + \)\(31\!\cdots\!84\)\( T^{12} - \)\(38\!\cdots\!60\)\( T^{14} + \)\(42\!\cdots\!14\)\( T^{16} - \)\(41\!\cdots\!08\)\( T^{18} + \)\(36\!\cdots\!64\)\( T^{20} - \)\(29\!\cdots\!08\)\( T^{22} + \)\(21\!\cdots\!14\)\( T^{24} - \)\(13\!\cdots\!60\)\( T^{26} + \)\(81\!\cdots\!84\)\( T^{28} - \)\(41\!\cdots\!48\)\( T^{30} + \)\(17\!\cdots\!17\)\( T^{32} - \)\(63\!\cdots\!00\)\( T^{34} + \)\(17\!\cdots\!66\)\( T^{36} - \)\(34\!\cdots\!48\)\( T^{38} + \)\(34\!\cdots\!01\)\( T^{40} \)
$67$ \( 1 - 11828518964 T^{2} + 68669252417166303634 T^{4} - \)\(26\!\cdots\!60\)\( T^{6} + \)\(76\!\cdots\!57\)\( T^{8} - \)\(18\!\cdots\!04\)\( T^{10} + \)\(38\!\cdots\!16\)\( T^{12} - \)\(70\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!54\)\( T^{16} - \)\(18\!\cdots\!64\)\( T^{18} + \)\(25\!\cdots\!76\)\( T^{20} - \)\(33\!\cdots\!36\)\( T^{22} + \)\(39\!\cdots\!54\)\( T^{24} - \)\(42\!\cdots\!60\)\( T^{26} + \)\(42\!\cdots\!16\)\( T^{28} - \)\(36\!\cdots\!96\)\( T^{30} + \)\(28\!\cdots\!57\)\( T^{32} - \)\(17\!\cdots\!40\)\( T^{34} + \)\(83\!\cdots\!34\)\( T^{36} - \)\(26\!\cdots\!36\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \)
$71$ \( ( 1 - 100156 T + 13448448446 T^{2} - 957445823975748 T^{3} + 76873855601451932317 T^{4} - \)\(43\!\cdots\!20\)\( T^{5} + \)\(26\!\cdots\!28\)\( T^{6} - \)\(12\!\cdots\!92\)\( T^{7} + \)\(67\!\cdots\!74\)\( T^{8} - \)\(28\!\cdots\!84\)\( T^{9} + \)\(13\!\cdots\!68\)\( T^{10} - \)\(51\!\cdots\!84\)\( T^{11} + \)\(21\!\cdots\!74\)\( T^{12} - \)\(74\!\cdots\!92\)\( T^{13} + \)\(28\!\cdots\!28\)\( T^{14} - \)\(82\!\cdots\!20\)\( T^{15} + \)\(26\!\cdots\!17\)\( T^{16} - \)\(59\!\cdots\!48\)\( T^{17} + \)\(15\!\cdots\!46\)\( T^{18} - \)\(20\!\cdots\!56\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$73$ \( ( 1 + 52568 T + 9379342894 T^{2} + 374528688123736 T^{3} + 37721970904921608509 T^{4} + \)\(11\!\cdots\!56\)\( T^{5} + \)\(82\!\cdots\!04\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!58\)\( T^{8} + \)\(24\!\cdots\!96\)\( T^{9} + \)\(16\!\cdots\!04\)\( T^{10} + \)\(51\!\cdots\!28\)\( T^{11} + \)\(49\!\cdots\!42\)\( T^{12} + \)\(17\!\cdots\!28\)\( T^{13} + \)\(15\!\cdots\!04\)\( T^{14} + \)\(44\!\cdots\!08\)\( T^{15} + \)\(29\!\cdots\!41\)\( T^{16} + \)\(61\!\cdots\!52\)\( T^{17} + \)\(31\!\cdots\!94\)\( T^{18} + \)\(37\!\cdots\!24\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} )^{2} \)
$79$ \( ( 1 + 141040 T + 30508439526 T^{2} + 3027236690742416 T^{3} + \)\(38\!\cdots\!93\)\( T^{4} + \)\(29\!\cdots\!88\)\( T^{5} + \)\(27\!\cdots\!48\)\( T^{6} + \)\(17\!\cdots\!16\)\( T^{7} + \)\(13\!\cdots\!30\)\( T^{8} + \)\(73\!\cdots\!40\)\( T^{9} + \)\(47\!\cdots\!04\)\( T^{10} + \)\(22\!\cdots\!60\)\( T^{11} + \)\(12\!\cdots\!30\)\( T^{12} + \)\(51\!\cdots\!84\)\( T^{13} + \)\(24\!\cdots\!48\)\( T^{14} + \)\(81\!\cdots\!12\)\( T^{15} + \)\(32\!\cdots\!93\)\( T^{16} + \)\(79\!\cdots\!84\)\( T^{17} + \)\(24\!\cdots\!26\)\( T^{18} + \)\(34\!\cdots\!60\)\( T^{19} + \)\(76\!\cdots\!01\)\( T^{20} )^{2} \)
$83$ \( 1 - 34768653380 T^{2} + \)\(55\!\cdots\!58\)\( T^{4} - \)\(53\!\cdots\!56\)\( T^{6} + \)\(34\!\cdots\!33\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{10} + \)\(44\!\cdots\!40\)\( T^{12} - \)\(24\!\cdots\!36\)\( T^{14} - \)\(64\!\cdots\!10\)\( T^{16} + \)\(49\!\cdots\!20\)\( T^{18} - \)\(23\!\cdots\!44\)\( T^{20} + \)\(76\!\cdots\!80\)\( T^{22} - \)\(15\!\cdots\!10\)\( T^{24} - \)\(90\!\cdots\!64\)\( T^{26} + \)\(25\!\cdots\!40\)\( T^{28} - \)\(14\!\cdots\!80\)\( T^{30} + \)\(48\!\cdots\!33\)\( T^{32} - \)\(11\!\cdots\!44\)\( T^{34} + \)\(18\!\cdots\!58\)\( T^{36} - \)\(18\!\cdots\!20\)\( T^{38} + \)\(80\!\cdots\!01\)\( T^{40} \)
$89$ \( ( 1 + 1580 T + 27398194046 T^{2} + 460040940498284 T^{3} + \)\(38\!\cdots\!93\)\( T^{4} + \)\(90\!\cdots\!72\)\( T^{5} + \)\(38\!\cdots\!08\)\( T^{6} + \)\(95\!\cdots\!24\)\( T^{7} + \)\(29\!\cdots\!70\)\( T^{8} + \)\(73\!\cdots\!60\)\( T^{9} + \)\(18\!\cdots\!64\)\( T^{10} + \)\(40\!\cdots\!40\)\( T^{11} + \)\(91\!\cdots\!70\)\( T^{12} + \)\(16\!\cdots\!76\)\( T^{13} + \)\(37\!\cdots\!08\)\( T^{14} + \)\(49\!\cdots\!28\)\( T^{15} + \)\(11\!\cdots\!93\)\( T^{16} + \)\(77\!\cdots\!16\)\( T^{17} + \)\(25\!\cdots\!46\)\( T^{18} + \)\(83\!\cdots\!20\)\( T^{19} + \)\(29\!\cdots\!01\)\( T^{20} )^{2} \)
$97$ \( ( 1 - 73688 T + 43672916862 T^{2} - 2817316775448856 T^{3} + \)\(98\!\cdots\!25\)\( T^{4} - \)\(59\!\cdots\!28\)\( T^{5} + \)\(15\!\cdots\!52\)\( T^{6} - \)\(87\!\cdots\!16\)\( T^{7} + \)\(18\!\cdots\!70\)\( T^{8} - \)\(96\!\cdots\!88\)\( T^{9} + \)\(17\!\cdots\!72\)\( T^{10} - \)\(82\!\cdots\!16\)\( T^{11} + \)\(13\!\cdots\!30\)\( T^{12} - \)\(55\!\cdots\!88\)\( T^{13} + \)\(84\!\cdots\!52\)\( T^{14} - \)\(27\!\cdots\!96\)\( T^{15} + \)\(39\!\cdots\!25\)\( T^{16} - \)\(97\!\cdots\!08\)\( T^{17} + \)\(12\!\cdots\!62\)\( T^{18} - \)\(18\!\cdots\!16\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} )^{2} \)
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