# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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