Properties

Label 6040.2.a.p
Level $6040$
Weight $2$
Character orbit 6040.a
Self dual yes
Analytic conductor $48.230$
Analytic rank $1$
Dimension $19$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6040.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2296428209\)
Analytic rank: \(1\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Defining polynomial: \(x^{19} - 5 x^{18} - 29 x^{17} + 165 x^{16} + 325 x^{15} - 2208 x^{14} - 1891 x^{13} + 15895 x^{12} + 6652 x^{11} - 67665 x^{10} - 17345 x^{9} + 174105 x^{8} + 41499 x^{7} - 262172 x^{6} - 80919 x^{5} + 206783 x^{4} + 91643 x^{3} - 59750 x^{2} - 40224 x - 5628\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + q^{5} + \beta_{3} q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + q^{5} + \beta_{3} q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} + ( -1 - \beta_{18} ) q^{11} + ( \beta_{1} - \beta_{14} ) q^{13} -\beta_{1} q^{15} -\beta_{13} q^{17} + ( \beta_{3} + \beta_{8} + \beta_{12} - \beta_{17} ) q^{19} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{9} ) q^{21} + ( -3 + \beta_{1} - 2 \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{17} + \beta_{18} ) q^{23} + q^{25} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{14} + \beta_{15} + \beta_{16} ) q^{27} + ( -2 - \beta_{2} - \beta_{3} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{29} + ( -1 - \beta_{1} - \beta_{8} - \beta_{16} + \beta_{18} ) q^{31} + ( \beta_{1} - \beta_{3} + \beta_{9} + \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{18} ) q^{33} + \beta_{3} q^{35} + ( -\beta_{2} + \beta_{3} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} ) q^{37} + ( -5 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{17} + \beta_{18} ) q^{39} + ( -1 - \beta_{3} - \beta_{5} + \beta_{7} - \beta_{11} + \beta_{14} + \beta_{18} ) q^{41} + ( -\beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{12} - \beta_{17} ) q^{43} + ( 1 + \beta_{1} + \beta_{2} ) q^{45} + ( \beta_{1} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{14} - \beta_{15} - \beta_{18} ) q^{47} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{16} ) q^{49} + ( 1 - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + \beta_{12} + 2 \beta_{16} - \beta_{17} ) q^{51} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} + 2 \beta_{7} - 2 \beta_{9} + \beta_{11} + \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{53} + ( -1 - \beta_{18} ) q^{55} + ( -1 + 2 \beta_{1} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{18} ) q^{57} + ( -3 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{17} ) q^{59} + ( -5 + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{16} ) q^{61} + ( -3 + \beta_{1} + \beta_{4} - \beta_{5} + 3 \beta_{7} - 2 \beta_{9} + \beta_{11} - 2 \beta_{16} + \beta_{18} ) q^{63} + ( \beta_{1} - \beta_{14} ) q^{65} + ( -\beta_{1} + 2 \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - 2 \beta_{17} - 2 \beta_{18} ) q^{67} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{12} - 2 \beta_{14} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{69} + ( -5 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{11} - 2 \beta_{12} + 2 \beta_{17} + \beta_{18} ) q^{71} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{73} -\beta_{1} q^{75} + ( -3 + \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{14} + 2 \beta_{15} + \beta_{17} + \beta_{18} ) q^{77} + ( -5 + 2 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 3 \beta_{9} - \beta_{10} + 4 \beta_{12} + 2 \beta_{13} + \beta_{15} + \beta_{16} ) q^{79} + ( 6 + 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{81} + ( -6 + 3 \beta_{1} - 4 \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{7} - \beta_{8} - 4 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{83} -\beta_{13} q^{85} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{7} - \beta_{8} - 4 \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + \beta_{15} - 4 \beta_{16} + 2 \beta_{17} + \beta_{18} ) q^{87} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{15} + 2 \beta_{16} - \beta_{17} - \beta_{18} ) q^{89} + ( 2 \beta_{2} + 3 \beta_{3} - \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} - \beta_{17} ) q^{91} + ( 3 + \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} - 2 \beta_{18} ) q^{93} + ( \beta_{3} + \beta_{8} + \beta_{12} - \beta_{17} ) q^{95} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} + 2 \beta_{14} + \beta_{15} + \beta_{18} ) q^{97} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{13} - 3 \beta_{14} - \beta_{15} - \beta_{17} - 4 \beta_{18} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19q - 5q^{3} + 19q^{5} - 8q^{7} + 26q^{9} + O(q^{10}) \) \( 19q - 5q^{3} + 19q^{5} - 8q^{7} + 26q^{9} - 18q^{11} + 5q^{13} - 5q^{15} - 4q^{17} - 27q^{19} - 18q^{21} - 25q^{23} + 19q^{25} - 35q^{27} - 35q^{29} - 26q^{31} - 8q^{35} - 10q^{37} - 48q^{39} - 14q^{41} - 21q^{43} + 26q^{45} - 40q^{47} + 23q^{49} - 32q^{51} - 3q^{53} - 18q^{55} - 13q^{57} - 28q^{59} - 46q^{61} - 53q^{63} + 5q^{65} - 42q^{67} - 31q^{69} - 46q^{71} + 31q^{73} - 5q^{75} + 15q^{77} - 56q^{79} + 31q^{81} - 25q^{83} - 4q^{85} - 20q^{87} - 7q^{89} - 61q^{91} + 29q^{93} - 27q^{95} + 39q^{97} - 52q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19} - 5 x^{18} - 29 x^{17} + 165 x^{16} + 325 x^{15} - 2208 x^{14} - 1891 x^{13} + 15895 x^{12} + 6652 x^{11} - 67665 x^{10} - 17345 x^{9} + 174105 x^{8} + 41499 x^{7} - 262172 x^{6} - 80919 x^{5} + 206783 x^{4} + 91643 x^{3} - 59750 x^{2} - 40224 x - 5628\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\(55395572792623 \nu^{18} - 1282570164104816 \nu^{17} + 3843828599512745 \nu^{16} + 33567815192362180 \nu^{15} - 144066698105804193 \nu^{14} - 325977195535059477 \nu^{13} + 1792759675120172710 \nu^{12} + 1588052770850214607 \nu^{11} - 11294005761518533393 \nu^{10} - 4500209660384611576 \nu^{9} + 40067118650304619217 \nu^{8} + 8934620047045302660 \nu^{7} - 80173003627501509915 \nu^{6} - 14636171244761239999 \nu^{5} + 82631139058765852268 \nu^{4} + 16339925852923382945 \nu^{3} - 33212601804754250690 \nu^{2} - 7576366659179760236 \nu + 1011634103784965924\)\()/ 228771461625160096 \)
\(\beta_{4}\)\(=\)\((\)\(-825471425164009 \nu^{18} + 5148768030504856 \nu^{17} + 16656356771269549 \nu^{16} - 150533195307394516 \nu^{15} - 67958043505442033 \nu^{14} + 1739613051424907459 \nu^{13} - 567866092541121870 \nu^{12} - 10775530281108362853 \nu^{11} + 6541382809435982627 \nu^{10} + 39992548162303710512 \nu^{9} - 27241571945505753767 \nu^{8} - 91839653932711203196 \nu^{7} + 58655996588257041457 \nu^{6} + 126453980894546361929 \nu^{5} - 62145344231637892636 \nu^{4} - 93399395100110890643 \nu^{3} + 20011205006996581446 \nu^{2} + 28137440905699534180 \nu + 3451667624814453428\)\()/ 457542923250320192 \)
\(\beta_{5}\)\(=\)\((\)\(831149248013573 \nu^{18} - 6529221237850736 \nu^{17} - 8468181688802937 \nu^{16} + 172205964148202772 \nu^{15} - 143534913928110307 \nu^{14} - 1694597100662116159 \nu^{13} + 2487208350510041006 \nu^{12} + 8381720526370062953 \nu^{11} - 14128580590427639255 \nu^{10} - 23375799369623611568 \nu^{9} + 37623785569757123579 \nu^{8} + 39546479291651782084 \nu^{7} - 47987975251419114109 \nu^{6} - 43115468783407000837 \nu^{5} + 27556736624880712228 \nu^{4} + 28276765452349644391 \nu^{3} - 9151291982405240270 \nu^{2} - 6077448197860154164 \nu + 1614654699627908572\)\()/ 457542923250320192 \)
\(\beta_{6}\)\(=\)\((\)\(893262017742841 \nu^{18} - 6709911317038048 \nu^{17} - 4374937933861837 \nu^{16} + 143581904507289700 \nu^{15} - 236985751269460111 \nu^{14} - 857864446451730955 \nu^{13} + 2836792745171684454 \nu^{12} - 100863235120512307 \nu^{11} - 11438341612028607971 \nu^{10} + 18713949331749328624 \nu^{9} + 13366198949241451367 \nu^{8} - 72912510979539031644 \nu^{7} + 23448483493090406239 \nu^{6} + 119709595019567528583 \nu^{5} - 64647877824025594908 \nu^{4} - 91149929208499445773 \nu^{3} + 34787223627735288026 \nu^{2} + 29631649698934677180 \nu + 1722308039210366348\)\()/ 457542923250320192 \)
\(\beta_{7}\)\(=\)\((\)\(56636909905056 \nu^{18} - 466859136090342 \nu^{17} - 472631481848911 \nu^{16} + 12662979091017916 \nu^{15} - 14351726530935302 \nu^{14} - 129569857849845676 \nu^{13} + 242072346115647165 \nu^{12} + 676539570406064409 \nu^{11} - 1504470703145431741 \nu^{10} - 2096524180851425584 \nu^{9} + 4638648378150119924 \nu^{8} + 4537703098655126976 \nu^{7} - 7232763185988054269 \nu^{6} - 7699778770407449598 \nu^{5} + 4793176495819915334 \nu^{4} + 8624873138914767615 \nu^{3} - 30204773087751264 \nu^{2} - 3889761727382701440 \nu - 982608926580490792\)\()/ 28596432703145012 \)
\(\beta_{8}\)\(=\)\((\)\(-1842573741871917 \nu^{18} + 7487676163763712 \nu^{17} + 56247718798349417 \nu^{16} - 234583820154108356 \nu^{15} - 689632753169362133 \nu^{14} + 2894698946932575287 \nu^{13} + 4616023233902091338 \nu^{12} - 18426961314159695961 \nu^{11} - 19088000936660001737 \nu^{10} + 65042439574070369888 \nu^{9} + 51035187199415557773 \nu^{8} - 124009163196399080772 \nu^{7} - 85870096282012514947 \nu^{6} + 107127077210264354573 \nu^{5} + 80227629513941781164 \nu^{4} - 9135263353863909511 \nu^{3} - 27312540334604478130 \nu^{2} - 25973098539530318860 \nu - 6696186093159290140\)\()/ 457542923250320192 \)
\(\beta_{9}\)\(=\)\((\)\(-265246968233581 \nu^{18} + 1683217593650907 \nu^{17} + 5145929270516660 \nu^{16} - 48909518613942212 \nu^{15} - 14899268175785925 \nu^{14} + 555872474701520570 \nu^{13} - 271304883452175172 \nu^{12} - 3312637470646319049 \nu^{11} + 2635534383536689278 \nu^{10} + 11382041130194319182 \nu^{9} - 10194286201079735993 \nu^{8} - 22851621137466968613 \nu^{7} + 20015000123231956768 \nu^{6} + 25437466164583338201 \nu^{5} - 19436518983654857783 \nu^{4} - 13657286033777995806 \nu^{3} + 7179337289577138652 \nu^{2} + 2818443801556378476 \nu - 60581224214440872\)\()/ 57192865406290024 \)
\(\beta_{10}\)\(=\)\((\)\(2144199181458703 \nu^{18} - 14007918034232216 \nu^{17} - 38711864397166811 \nu^{16} + 401669051946076876 \nu^{15} + 46336575550144007 \nu^{14} - 4507542403554445333 \nu^{13} + 2875138811829736290 \nu^{12} + 26864902225484161459 \nu^{11} - 24123074281514233269 \nu^{10} - 95642613213777855920 \nu^{9} + 86611245129153271489 \nu^{8} + 213623456478839512564 \nu^{7} - 155286643777086364727 \nu^{6} - 297154724962151344431 \nu^{5} + 124888418158671406260 \nu^{4} + 233698178817215866373 \nu^{3} - 18400403424873851690 \nu^{2} - 77031313802425172604 \nu - 16633657644450319404\)\()/ 457542923250320192 \)
\(\beta_{11}\)\(=\)\((\)\(-2543053966709621 \nu^{18} + 19553203998112072 \nu^{17} + 30277054135873193 \nu^{16} - 548476037235782756 \nu^{15} + 419594042994086787 \nu^{14} + 5903216243609510583 \nu^{13} - 9019156893641800262 \nu^{12} - 32741676166555832273 \nu^{11} + 62547489597013713767 \nu^{10} + 104760723853281959600 \nu^{9} - 217003369085973033403 \nu^{8} - 206900403791732026524 \nu^{7} + 399813773255867433917 \nu^{6} + 267992302397578881109 \nu^{5} - 363351139346577945276 \nu^{4} - 225766875158450189399 \nu^{3} + 116871191622166077710 \nu^{2} + 89834398748893076436 \nu + 11925598053551983396\)\()/ 457542923250320192 \)
\(\beta_{12}\)\(=\)\((\)\(-648499956243255 \nu^{18} + 5012064203537208 \nu^{17} + 8667195040728045 \nu^{16} - 145923623282507952 \nu^{15} + 84746839523124309 \nu^{14} + 1666477337278914385 \nu^{13} - 2131813093126046692 \nu^{12} - 10077000035553477193 \nu^{11} + 15555986639844284415 \nu^{10} + 36144995093235280244 \nu^{9} - 56032071984249745465 \nu^{8} - 80756436732202726424 \nu^{7} + 106048900728093493181 \nu^{6} + 112635352391755816023 \nu^{5} - 96314212157456324132 \nu^{4} - 90911545845393234887 \nu^{3} + 26840551255631058438 \nu^{2} + 32005685014482075124 \nu + 5986425026293859236\)\()/ 114385730812580048 \)
\(\beta_{13}\)\(=\)\((\)\(-1374745276971833 \nu^{18} + 6233607352857088 \nu^{17} + 40860651477333453 \nu^{16} - 201481782286309668 \nu^{15} - 474806648528782897 \nu^{14} + 2607313444268021163 \nu^{13} + 2870798724368680730 \nu^{12} - 17811058676487844173 \nu^{11} - 9974466268086971869 \nu^{10} + 69983881375983107312 \nu^{9} + 20776420739350952313 \nu^{8} - 159794259178220365380 \nu^{7} - 27366939526710095807 \nu^{6} + 202499225907393933433 \nu^{5} + 27096772152993291772 \nu^{4} - 126908161268216007379 \nu^{3} - 22574056397359268538 \nu^{2} + 30638708633486468964 \nu + 8167086807839149108\)\()/ 228771461625160096 \)
\(\beta_{14}\)\(=\)\((\)\(-3039888107633601 \nu^{18} + 21385926549656952 \nu^{17} + 43569315816672797 \nu^{16} - 584514148862704228 \nu^{15} + 237007316844629479 \nu^{14} + 6020226093986778971 \nu^{13} - 6978674237471914710 \nu^{12} - 31024857777623091109 \nu^{11} + 46387313396762744579 \nu^{10} + 87392079449956939424 \nu^{9} - 141094929426519309839 \nu^{8} - 138163079917435060844 \nu^{7} + 205522033580513668913 \nu^{6} + 127604654796804573089 \nu^{5} - 121490766031181554524 \nu^{4} - 77599544991083627219 \nu^{3} + 11835540468138859814 \nu^{2} + 24582797901195324452 \nu + 6670908614249021236\)\()/ 457542923250320192 \)
\(\beta_{15}\)\(=\)\((\)\(1652069990496175 \nu^{18} - 12540283023594488 \nu^{17} - 21564673027731835 \nu^{16} + 357014723638780620 \nu^{15} - 219304735752279001 \nu^{14} - 3937975463808284437 \nu^{13} + 5278789286287108706 \nu^{12} + 22691217218032930995 \nu^{11} - 37297691424213731637 \nu^{10} - 76764217475065407952 \nu^{9} + 129528865730130573569 \nu^{8} + 162421707726185401652 \nu^{7} - 235389831702239684215 \nu^{6} - 221308435501997123887 \nu^{5} + 205953568009256349140 \nu^{4} + 182585464307163511781 \nu^{3} - 58440175368525168138 \nu^{2} - 65490892598179360316 \nu - 10388116111708522412\)\()/ 228771461625160096 \)
\(\beta_{16}\)\(=\)\((\)\(4125642740291903 \nu^{18} - 27103994474524504 \nu^{17} - 75818776899204467 \nu^{16} + 789591293776478876 \nu^{15} + 86357606179246823 \nu^{14} - 8966005037752738805 \nu^{13} + 6088557687263850570 \nu^{12} + 53110027235529915035 \nu^{11} - 52220481059478395469 \nu^{10} - 181821342301299894016 \nu^{9} + 191189145105819027985 \nu^{8} + 373141090695751187588 \nu^{7} - 349667716662878613727 \nu^{6} - 457820635907237577503 \nu^{5} + 296565179915255069556 \nu^{4} + 316755291677928609789 \nu^{3} - 72028452155976973882 \nu^{2} - 97455190035628303772 \nu - 17922009821745376204\)\()/ 457542923250320192 \)
\(\beta_{17}\)\(=\)\((\)\(-5734380684972101 \nu^{18} + 32134720242086384 \nu^{17} + 141604639245533105 \nu^{16} - 1005499581532864900 \nu^{15} - 1114941497406725869 \nu^{14} + 12534591592790226543 \nu^{13} + 1861805423561134778 \nu^{12} - 82659828787906528913 \nu^{11} + 18907168699394828847 \nu^{10} + 316724397005442684960 \nu^{9} - 114197716146189228923 \nu^{8} - 719721317464493735924 \nu^{7} + 254723974714131028581 \nu^{6} + 939373349287456002501 \nu^{5} - 234702087701587701668 \nu^{4} - 638524446302924062879 \nu^{3} + 39594315574452729118 \nu^{2} + 175451898025721634132 \nu + 33803425560306121028\)\()/ 457542923250320192 \)
\(\beta_{18}\)\(=\)\((\)\(6000740936115901 \nu^{18} - 37872303245196736 \nu^{17} - 117375998014468801 \nu^{16} + 1104923052372439220 \nu^{15} + 346768879365177429 \nu^{14} - 12588330322487702519 \nu^{13} + 6177222909748312958 \nu^{12} + 74989229416502241633 \nu^{11} - 59542132916546240239 \nu^{10} - 258328728461468122512 \nu^{9} + 223294169369261280803 \nu^{8} + 530761454648421512276 \nu^{7} - 410391423955509687781 \nu^{6} - 641770077230588731133 \nu^{5} + 348640318779203430292 \nu^{4} + 426181604695001605439 \nu^{3} - 88184304972892466494 \nu^{2} - 122429703947055203924 \nu - 19426824517150598020\)\()/ 457542923250320192 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{16} - \beta_{15} - \beta_{14} + \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + 7 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-2 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 4 \beta_{3} + 12 \beta_{2} + 11 \beta_{1} + 33\)
\(\nu^{5}\)\(=\)\(\beta_{17} - 17 \beta_{16} - 12 \beta_{15} - 14 \beta_{14} - \beta_{13} - 3 \beta_{12} + 2 \beta_{11} + 11 \beta_{10} - 16 \beta_{9} - 3 \beta_{8} + 17 \beta_{7} - 14 \beta_{6} + 14 \beta_{3} + 15 \beta_{2} + 61 \beta_{1} + 34\)
\(\nu^{6}\)\(=\)\(-3 \beta_{18} - \beta_{17} + 4 \beta_{15} - 32 \beta_{14} - 34 \beta_{13} + 32 \beta_{12} + 2 \beta_{11} + 10 \beta_{10} + 28 \beta_{9} + 32 \beta_{8} - 29 \beta_{7} - 13 \beta_{6} + 13 \beta_{5} - 16 \beta_{4} + 65 \beta_{3} + 135 \beta_{2} + 114 \beta_{1} + 323\)
\(\nu^{7}\)\(=\)\(-5 \beta_{18} + 14 \beta_{17} - 222 \beta_{16} - 128 \beta_{15} - 174 \beta_{14} - 22 \beta_{13} - 46 \beta_{12} + 33 \beta_{11} + 112 \beta_{10} - 204 \beta_{9} - 46 \beta_{8} + 216 \beta_{7} - 166 \beta_{6} - 5 \beta_{5} - 5 \beta_{4} + 172 \beta_{3} + 178 \beta_{2} + 597 \beta_{1} + 351\)
\(\nu^{8}\)\(=\)\(-64 \beta_{18} - 25 \beta_{17} - 6 \beta_{16} + 92 \beta_{15} - 411 \beta_{14} - 444 \beta_{13} + 417 \beta_{12} + 41 \beta_{11} + 81 \beta_{10} + 317 \beta_{9} + 411 \beta_{8} - 348 \beta_{7} - 142 \beta_{6} + 133 \beta_{5} - 210 \beta_{4} + 829 \beta_{3} + 1509 \beta_{2} + 1189 \beta_{1} + 3380\)
\(\nu^{9}\)\(=\)\(-121 \beta_{18} + 146 \beta_{17} - 2662 \beta_{16} - 1350 \beta_{15} - 2074 \beta_{14} - 321 \beta_{13} - 541 \beta_{12} + 427 \beta_{11} + 1161 \beta_{10} - 2439 \beta_{9} - 548 \beta_{8} + 2534 \beta_{7} - 1887 \beta_{6} - 131 \beta_{5} - 123 \beta_{4} + 2018 \beta_{3} + 1988 \beta_{2} + 6212 \beta_{1} + 3583\)
\(\nu^{10}\)\(=\)\(-959 \beta_{18} - 409 \beta_{17} - 118 \beta_{16} + 1463 \beta_{15} - 4922 \beta_{14} - 5324 \beta_{13} + 5077 \beta_{12} + 603 \beta_{11} + 621 \beta_{10} + 3470 \beta_{9} + 4940 \beta_{8} - 4038 \beta_{7} - 1498 \beta_{6} + 1272 \beta_{5} - 2610 \beta_{4} + 9788 \beta_{3} + 16847 \beta_{2} + 12560 \beta_{1} + 36456\)
\(\nu^{11}\)\(=\)\(-1920 \beta_{18} + 1412 \beta_{17} - 30868 \beta_{16} - 14337 \beta_{15} - 24113 \beta_{14} - 3992 \beta_{13} - 5950 \beta_{12} + 5170 \beta_{11} + 12315 \beta_{10} - 28447 \beta_{9} - 6145 \beta_{8} + 28984 \beta_{7} - 21179 \beta_{6} - 2293 \beta_{5} - 2047 \beta_{4} + 23051 \beta_{3} + 21755 \beta_{2} + 66682 \beta_{1} + 36668\)
\(\nu^{12}\)\(=\)\(-12470 \beta_{18} - 5672 \beta_{17} - 1427 \beta_{16} + 20174 \beta_{15} - 57021 \beta_{14} - 61588 \beta_{13} + 59950 \beta_{12} + 7862 \beta_{11} + 4546 \beta_{10} + 38202 \beta_{9} + 57796 \beta_{8} - 46614 \beta_{7} - 15628 \beta_{6} + 11930 \beta_{5} - 31564 \beta_{4} + 111958 \beta_{3} + 188003 \beta_{2} + 134116 \beta_{1} + 399005\)
\(\nu^{13}\)\(=\)\(-25505 \beta_{18} + 13564 \beta_{17} - 352686 \beta_{16} - 153886 \beta_{15} - 275852 \beta_{14} - 45889 \beta_{13} - 64801 \beta_{12} + 61073 \beta_{11} + 132740 \beta_{10} - 327797 \beta_{9} - 68239 \beta_{8} + 328790 \beta_{7} - 236702 \beta_{6} - 33859 \beta_{5} - 28878 \beta_{4} + 259007 \beta_{3} + 236089 \beta_{2} + 727192 \beta_{1} + 376952\)
\(\nu^{14}\)\(=\)\(-150682 \beta_{18} - 72638 \beta_{17} - 12258 \beta_{16} + 259294 \beta_{15} - 648026 \beta_{14} - 700250 \beta_{13} + 696622 \beta_{12} + 96978 \beta_{11} + 29927 \beta_{10} + 426198 \beta_{9} + 667481 \beta_{8} - 537800 \beta_{7} - 161710 \beta_{6} + 111779 \beta_{5} - 374403 \beta_{4} + 1262091 \beta_{3} + 2097637 \beta_{2} + 1443169 \beta_{1} + 4400719\)
\(\nu^{15}\)\(=\)\(-308282 \beta_{18} + 132598 \beta_{17} - 4000805 \beta_{16} - 1669352 \beta_{15} - 3122385 \beta_{14} - 504470 \beta_{13} - 711853 \beta_{12} + 712758 \beta_{11} + 1445104 \beta_{10} - 3750518 \beta_{9} - 761410 \beta_{8} + 3720259 \beta_{7} - 2641192 \beta_{6} - 456732 \beta_{5} - 372564 \beta_{4} + 2881845 \beta_{3} + 2551366 \beta_{2} + 7995998 \beta_{1} + 3889559\)
\(\nu^{16}\)\(=\)\(-1743576 \beta_{18} - 891156 \beta_{17} - 53088 \beta_{16} + 3203127 \beta_{15} - 7274769 \beta_{14} - 7890282 \beta_{13} + 8019421 \beta_{12} + 1161585 \beta_{11} + 140965 \beta_{10} + 4814026 \beta_{9} + 7654625 \beta_{8} - 6202090 \beta_{7} - 1657891 \beta_{6} + 1055320 \beta_{5} - 4375233 \beta_{4} + 14127675 \beta_{3} + 23403975 \beta_{2} + 15608893 \beta_{1} + 48752267\)
\(\nu^{17}\)\(=\)\(-3518378 \beta_{18} + 1327790 \beta_{17} - 45212269 \beta_{16} - 18279525 \beta_{15} - 35092958 \beta_{14} - 5388501 \beta_{13} - 7921385 \beta_{12} + 8256408 \beta_{11} + 15829354 \beta_{10} - 42713866 \beta_{9} - 8561836 \beta_{8} + 42071022 \beta_{7} - 29449614 \beta_{6} - 5838612 \beta_{5} - 4550413 \beta_{4} + 31882688 \beta_{3} + 27500351 \beta_{2} + 88319490 \beta_{1} + 40226199\)
\(\nu^{18}\)\(=\)\(-19618273 \beta_{18} - 10663514 \beta_{17} + 691708 \beta_{16} + 38617512 \beta_{15} - 80990164 \beta_{14} - 88464006 \beta_{13} + 91778379 \beta_{12} + 13673570 \beta_{11} - 318919 \beta_{10} + 54898170 \beta_{9} + 87414312 \beta_{8} - 71469117 \beta_{7} - 16809780 \beta_{6} + 10083123 \beta_{5} - 50542513 \beta_{4} + 157595999 \beta_{3} + 261149835 \beta_{2} + 169370911 \beta_{1} + 541608246\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.33655
3.31016
2.89238
2.44352
2.34269
1.90812
1.68584
1.40330
1.02734
−0.237601
−0.435585
−0.815820
−0.942024
−1.62033
−1.76710
−1.97307
−2.06929
−2.11583
−3.37325
0 −3.33655 0 1.00000 0 2.35293 0 8.13258 0
1.2 0 −3.31016 0 1.00000 0 −0.477457 0 7.95713 0
1.3 0 −2.89238 0 1.00000 0 0.451178 0 5.36583 0
1.4 0 −2.44352 0 1.00000 0 −4.90700 0 2.97081 0
1.5 0 −2.34269 0 1.00000 0 −1.53409 0 2.48818 0
1.6 0 −1.90812 0 1.00000 0 1.15158 0 0.640908 0
1.7 0 −1.68584 0 1.00000 0 −4.78926 0 −0.157936 0
1.8 0 −1.40330 0 1.00000 0 3.27194 0 −1.03074 0
1.9 0 −1.02734 0 1.00000 0 3.96391 0 −1.94457 0
1.10 0 0.237601 0 1.00000 0 4.27350 0 −2.94355 0
1.11 0 0.435585 0 1.00000 0 −2.87898 0 −2.81027 0
1.12 0 0.815820 0 1.00000 0 −2.49382 0 −2.33444 0
1.13 0 0.942024 0 1.00000 0 1.85968 0 −2.11259 0
1.14 0 1.62033 0 1.00000 0 1.05807 0 −0.374546 0
1.15 0 1.76710 0 1.00000 0 0.0387940 0 0.122653 0
1.16 0 1.97307 0 1.00000 0 −0.188446 0 0.892989 0
1.17 0 2.06929 0 1.00000 0 −0.835530 0 1.28196 0
1.18 0 2.11583 0 1.00000 0 −4.51128 0 1.47674 0
1.19 0 3.37325 0 1.00000 0 −3.80572 0 8.37885 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(151\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6040.2.a.p 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6040.2.a.p 19 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\(T_{3}^{19} + \cdots\)
\(T_{7}^{19} + \cdots\)
\(T_{11}^{19} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 5 T + 28 T^{2} + 105 T^{3} + 385 T^{4} + 1173 T^{5} + 3401 T^{6} + 8801 T^{7} + 21544 T^{8} + 48297 T^{9} + 102298 T^{10} + 200259 T^{11} + 369639 T^{12} + 629345 T^{13} + 1007640 T^{14} + 1476676 T^{15} + 2050061 T^{16} + 2643230 T^{17} + 3589755 T^{18} + 5417718 T^{19} + 10769265 T^{20} + 23789070 T^{21} + 55351647 T^{22} + 119610756 T^{23} + 244856520 T^{24} + 458792505 T^{25} + 808400493 T^{26} + 1313899299 T^{27} + 2013531534 T^{28} + 2851889553 T^{29} + 3816454968 T^{30} + 4677212241 T^{31} + 5422292523 T^{32} + 5610422637 T^{33} + 5524329195 T^{34} + 4519905705 T^{35} + 3615924564 T^{36} + 1937102445 T^{37} + 1162261467 T^{38} \)
$5$ \( ( 1 - T )^{19} \)
$7$ \( 1 + 8 T + 87 T^{2} + 543 T^{3} + 3565 T^{4} + 18384 T^{5} + 93619 T^{6} + 415955 T^{7} + 1798485 T^{8} + 7097291 T^{9} + 27221401 T^{10} + 97470412 T^{11} + 339791807 T^{12} + 1119899050 T^{13} + 3599730515 T^{14} + 11020265087 T^{15} + 32941393184 T^{16} + 94173390003 T^{17} + 263041879420 T^{18} + 703999382598 T^{19} + 1841293155940 T^{20} + 4614496110147 T^{21} + 11298897862112 T^{22} + 26459656473887 T^{23} + 60500670765605 T^{24} + 131755003333450 T^{25} + 279833164112201 T^{26} + 561897528568012 T^{27} + 1098481717943407 T^{28} + 2004809042450459 T^{29} + 3556192487384355 T^{30} + 5757352617691955 T^{31} + 9070652265292933 T^{32} + 12468452971256016 T^{33} + 16925056782946795 T^{34} + 18045481299293343 T^{35} + 20238854716887009 T^{36} + 13027308783283592 T^{37} + 11398895185373143 T^{38} \)
$11$ \( 1 + 18 T + 247 T^{2} + 2454 T^{3} + 21181 T^{4} + 156509 T^{5} + 1054756 T^{6} + 6437200 T^{7} + 36707246 T^{8} + 194628472 T^{9} + 977384365 T^{10} + 4629917152 T^{11} + 20942197301 T^{12} + 90115730775 T^{13} + 372151751391 T^{14} + 1469604097856 T^{15} + 5586871636156 T^{16} + 20369777201669 T^{17} + 71617697359258 T^{18} + 241820759643478 T^{19} + 787794670951838 T^{20} + 2464743041401949 T^{21} + 7436126147723636 T^{22} + 21516473596709696 T^{23} + 59935411713271941 T^{24} + 159645514127489775 T^{25} + 408104179920325471 T^{26} + 992463859825426912 T^{27} + 2304621206671251215 T^{28} + 5048161317707839672 T^{29} + 10473005679788947306 T^{30} + 20202691146628421200 T^{31} + 36413037770084085836 T^{32} + 59434266704279465669 T^{33} + 88478293476392903831 T^{34} + \)\(11\!\cdots\!94\)\( T^{35} + \)\(12\!\cdots\!37\)\( T^{36} + \)\(10\!\cdots\!58\)\( T^{37} + 61159090448414546291 T^{38} \)
$13$ \( 1 - 5 T + 103 T^{2} - 415 T^{3} + 5372 T^{4} - 20237 T^{5} + 199251 T^{6} - 742795 T^{7} + 5808966 T^{8} - 21876562 T^{9} + 140058255 T^{10} - 538343803 T^{11} + 2894272436 T^{12} - 11321669092 T^{13} + 52360879452 T^{14} - 205835429828 T^{15} + 841516816285 T^{16} - 3263227299131 T^{17} + 12138647794523 T^{18} - 45310613127736 T^{19} + 157802421328799 T^{20} - 551485413553139 T^{21} + 1848812445378145 T^{22} - 5878865711317508 T^{23} + 19441228014371436 T^{24} - 54647534268287428 T^{25} + 181611303152977412 T^{26} - 439143578567071963 T^{27} + 1485247677330974115 T^{28} - 3015869844161143138 T^{29} + 10410598795507535742 T^{30} - 17305701138553274395 T^{31} + 60348167863613002503 T^{32} - 79680685917396511493 T^{33} + \)\(27\!\cdots\!04\)\( T^{34} - \)\(27\!\cdots\!15\)\( T^{35} + \)\(89\!\cdots\!99\)\( T^{36} - \)\(56\!\cdots\!45\)\( T^{37} + \)\(14\!\cdots\!77\)\( T^{38} \)
$17$ \( 1 + 4 T + 129 T^{2} + 555 T^{3} + 8680 T^{4} + 38720 T^{5} + 403087 T^{6} + 1830895 T^{7} + 14546768 T^{8} + 66293924 T^{9} + 437116547 T^{10} + 1967353042 T^{11} + 11423251269 T^{12} + 49921515276 T^{13} + 266126037332 T^{14} + 1112655834529 T^{15} + 5581247893596 T^{16} + 22133710651532 T^{17} + 105465619387167 T^{18} + 396183904285542 T^{19} + 1792915529581839 T^{20} + 6396642378292748 T^{21} + 27420670901237148 T^{22} + 92930127955696609 T^{23} + 377860916988101524 T^{24} + 1204984019559004044 T^{25} + 4687401767067026037 T^{26} + 13723777621805485522 T^{27} + 51836723090431095859 T^{28} + \)\(13\!\cdots\!76\)\( T^{29} + \)\(49\!\cdots\!44\)\( T^{30} + \)\(10\!\cdots\!95\)\( T^{31} + \)\(39\!\cdots\!19\)\( T^{32} + \)\(65\!\cdots\!80\)\( T^{33} + \)\(24\!\cdots\!40\)\( T^{34} + \)\(27\!\cdots\!55\)\( T^{35} + \)\(10\!\cdots\!33\)\( T^{36} + \)\(56\!\cdots\!36\)\( T^{37} + \)\(23\!\cdots\!53\)\( T^{38} \)
$19$ \( 1 + 27 T + 535 T^{2} + 7831 T^{3} + 97628 T^{4} + 1046136 T^{5} + 10062922 T^{6} + 87462094 T^{7} + 700547907 T^{8} + 5194046040 T^{9} + 36042332073 T^{10} + 234789044331 T^{11} + 1445917508080 T^{12} + 8435292346429 T^{13} + 46841230345474 T^{14} + 247896735734358 T^{15} + 1254523249513997 T^{16} + 6073818712825768 T^{17} + 28195688928536073 T^{18} + 125457254289174004 T^{19} + 535718089642185387 T^{20} + 2192648555330102248 T^{21} + 8604774968416505423 T^{22} + 32306150497637268918 T^{23} + \)\(11\!\cdots\!26\)\( T^{24} + \)\(39\!\cdots\!49\)\( T^{25} + \)\(12\!\cdots\!20\)\( T^{26} + \)\(39\!\cdots\!71\)\( T^{27} + \)\(11\!\cdots\!67\)\( T^{28} + \)\(31\!\cdots\!40\)\( T^{29} + \)\(81\!\cdots\!33\)\( T^{30} + \)\(19\!\cdots\!34\)\( T^{31} + \)\(42\!\cdots\!98\)\( T^{32} + \)\(83\!\cdots\!56\)\( T^{33} + \)\(14\!\cdots\!72\)\( T^{34} + \)\(22\!\cdots\!11\)\( T^{35} + \)\(29\!\cdots\!65\)\( T^{36} + \)\(28\!\cdots\!07\)\( T^{37} + \)\(19\!\cdots\!79\)\( T^{38} \)
$23$ \( 1 + 25 T + 483 T^{2} + 6657 T^{3} + 78929 T^{4} + 788614 T^{5} + 7089783 T^{6} + 56746078 T^{7} + 418193193 T^{8} + 2811738201 T^{9} + 17621204996 T^{10} + 101906218735 T^{11} + 553061386246 T^{12} + 2784525495916 T^{13} + 13243571803035 T^{14} + 58857507541326 T^{15} + 252296052575820 T^{16} + 1045218587025787 T^{17} + 4488879747048825 T^{18} + 20343701113662066 T^{19} + 103244234182122975 T^{20} + 552920632536641323 T^{21} + 3069686071690001940 T^{22} + 16470743767872209166 T^{23} + 85240170669461701005 T^{24} + \)\(41\!\cdots\!24\)\( T^{25} + \)\(18\!\cdots\!62\)\( T^{26} + \)\(79\!\cdots\!35\)\( T^{27} + \)\(31\!\cdots\!48\)\( T^{28} + \)\(11\!\cdots\!49\)\( T^{29} + \)\(39\!\cdots\!11\)\( T^{30} + \)\(12\!\cdots\!38\)\( T^{31} + \)\(35\!\cdots\!89\)\( T^{32} + \)\(91\!\cdots\!26\)\( T^{33} + \)\(21\!\cdots\!03\)\( T^{34} + \)\(40\!\cdots\!77\)\( T^{35} + \)\(68\!\cdots\!49\)\( T^{36} + \)\(81\!\cdots\!25\)\( T^{37} + \)\(74\!\cdots\!87\)\( T^{38} \)
$29$ \( 1 + 35 T + 756 T^{2} + 12298 T^{3} + 167884 T^{4} + 2006813 T^{5} + 21660828 T^{6} + 214507339 T^{7} + 1974741700 T^{8} + 17035008433 T^{9} + 138669684363 T^{10} + 1070096187532 T^{11} + 7860580603761 T^{12} + 55116418019285 T^{13} + 369831555544579 T^{14} + 2378774068521476 T^{15} + 14689079687025666 T^{16} + 87161365931198394 T^{17} + 497392064222251313 T^{18} + 2730581369496168420 T^{19} + 14424369862445288077 T^{20} + 73302708748137849354 T^{21} + \)\(35\!\cdots\!74\)\( T^{22} + \)\(16\!\cdots\!56\)\( T^{23} + \)\(75\!\cdots\!71\)\( T^{24} + \)\(32\!\cdots\!85\)\( T^{25} + \)\(13\!\cdots\!49\)\( T^{26} + \)\(53\!\cdots\!52\)\( T^{27} + \)\(20\!\cdots\!47\)\( T^{28} + \)\(71\!\cdots\!33\)\( T^{29} + \)\(24\!\cdots\!00\)\( T^{30} + \)\(75\!\cdots\!99\)\( T^{31} + \)\(22\!\cdots\!92\)\( T^{32} + \)\(59\!\cdots\!53\)\( T^{33} + \)\(14\!\cdots\!16\)\( T^{34} + \)\(30\!\cdots\!58\)\( T^{35} + \)\(54\!\cdots\!04\)\( T^{36} + \)\(73\!\cdots\!35\)\( T^{37} + \)\(61\!\cdots\!69\)\( T^{38} \)
$31$ \( 1 + 26 T + 505 T^{2} + 7507 T^{3} + 97022 T^{4} + 1106201 T^{5} + 11561741 T^{6} + 111535761 T^{7} + 1009177797 T^{8} + 8601913136 T^{9} + 69613352337 T^{10} + 536555748661 T^{11} + 3956081439850 T^{12} + 27961546276091 T^{13} + 189939104670201 T^{14} + 1241754799579918 T^{15} + 7824148963012716 T^{16} + 47555622396051041 T^{17} + 279011364426555775 T^{18} + 1580753440138227764 T^{19} + 8649352297223229025 T^{20} + 45700953122605050401 T^{21} + \)\(23\!\cdots\!56\)\( T^{22} + \)\(11\!\cdots\!78\)\( T^{23} + \)\(54\!\cdots\!51\)\( T^{24} + \)\(24\!\cdots\!71\)\( T^{25} + \)\(10\!\cdots\!50\)\( T^{26} + \)\(45\!\cdots\!01\)\( T^{27} + \)\(18\!\cdots\!27\)\( T^{28} + \)\(70\!\cdots\!36\)\( T^{29} + \)\(25\!\cdots\!07\)\( T^{30} + \)\(87\!\cdots\!21\)\( T^{31} + \)\(28\!\cdots\!31\)\( T^{32} + \)\(83\!\cdots\!21\)\( T^{33} + \)\(22\!\cdots\!22\)\( T^{34} + \)\(54\!\cdots\!67\)\( T^{35} + \)\(11\!\cdots\!55\)\( T^{36} + \)\(18\!\cdots\!66\)\( T^{37} + \)\(21\!\cdots\!71\)\( T^{38} \)
$37$ \( 1 + 10 T + 270 T^{2} + 2394 T^{3} + 37842 T^{4} + 314825 T^{5} + 3792127 T^{6} + 30123787 T^{7} + 303631628 T^{8} + 2309430713 T^{9} + 20449708618 T^{10} + 148909147614 T^{11} + 1194431191170 T^{12} + 8309492269427 T^{13} + 61540830059600 T^{14} + 408036887621621 T^{15} + 2825032416976248 T^{16} + 17821276585784641 T^{17} + 116274960721861956 T^{18} + 696464521225933360 T^{19} + 4302173546708892372 T^{20} + 24397327645939173529 T^{21} + \)\(14\!\cdots\!44\)\( T^{22} + \)\(76\!\cdots\!81\)\( T^{23} + \)\(42\!\cdots\!00\)\( T^{24} + \)\(21\!\cdots\!43\)\( T^{25} + \)\(11\!\cdots\!10\)\( T^{26} + \)\(52\!\cdots\!94\)\( T^{27} + \)\(26\!\cdots\!86\)\( T^{28} + \)\(11\!\cdots\!37\)\( T^{29} + \)\(54\!\cdots\!64\)\( T^{30} + \)\(19\!\cdots\!47\)\( T^{31} + \)\(92\!\cdots\!19\)\( T^{32} + \)\(28\!\cdots\!25\)\( T^{33} + \)\(12\!\cdots\!06\)\( T^{34} + \)\(29\!\cdots\!54\)\( T^{35} + \)\(12\!\cdots\!90\)\( T^{36} + \)\(16\!\cdots\!90\)\( T^{37} + \)\(62\!\cdots\!73\)\( T^{38} \)
$41$ \( 1 + 14 T + 424 T^{2} + 4604 T^{3} + 81835 T^{4} + 748985 T^{5} + 10179182 T^{6} + 82890787 T^{7} + 951974493 T^{8} + 7136986289 T^{9} + 72632117051 T^{10} + 511503471494 T^{11} + 4731577791669 T^{12} + 31643979398651 T^{13} + 270180953787955 T^{14} + 1723542871284813 T^{15} + 13724585484282873 T^{16} + 83506658146740621 T^{17} + 625184036814135869 T^{18} + 3617718144565975516 T^{19} + 25632545509379570629 T^{20} + \)\(14\!\cdots\!01\)\( T^{21} + \)\(94\!\cdots\!33\)\( T^{22} + \)\(48\!\cdots\!93\)\( T^{23} + \)\(31\!\cdots\!55\)\( T^{24} + \)\(15\!\cdots\!91\)\( T^{25} + \)\(92\!\cdots\!89\)\( T^{26} + \)\(40\!\cdots\!74\)\( T^{27} + \)\(23\!\cdots\!11\)\( T^{28} + \)\(95\!\cdots\!89\)\( T^{29} + \)\(52\!\cdots\!13\)\( T^{30} + \)\(18\!\cdots\!47\)\( T^{31} + \)\(94\!\cdots\!22\)\( T^{32} + \)\(28\!\cdots\!85\)\( T^{33} + \)\(12\!\cdots\!35\)\( T^{34} + \)\(29\!\cdots\!64\)\( T^{35} + \)\(11\!\cdots\!44\)\( T^{36} + \)\(15\!\cdots\!94\)\( T^{37} + \)\(43\!\cdots\!61\)\( T^{38} \)
$43$ \( 1 + 21 T + 596 T^{2} + 9234 T^{3} + 161038 T^{4} + 2053315 T^{5} + 27773232 T^{6} + 306654655 T^{7} + 3499525009 T^{8} + 34439534170 T^{9} + 345679890922 T^{10} + 3086573198150 T^{11} + 27903332082978 T^{12} + 228654724550094 T^{13} + 1889108123082193 T^{14} + 14311447324461569 T^{15} + 109051257094336819 T^{16} + 767083715981138168 T^{17} + 5420618127619235826 T^{18} + 35470483556462799472 T^{19} + \)\(23\!\cdots\!18\)\( T^{20} + \)\(14\!\cdots\!32\)\( T^{21} + \)\(86\!\cdots\!33\)\( T^{22} + \)\(48\!\cdots\!69\)\( T^{23} + \)\(27\!\cdots\!99\)\( T^{24} + \)\(14\!\cdots\!06\)\( T^{25} + \)\(75\!\cdots\!46\)\( T^{26} + \)\(36\!\cdots\!50\)\( T^{27} + \)\(17\!\cdots\!46\)\( T^{28} + \)\(74\!\cdots\!30\)\( T^{29} + \)\(32\!\cdots\!63\)\( T^{30} + \)\(12\!\cdots\!55\)\( T^{31} + \)\(47\!\cdots\!76\)\( T^{32} + \)\(15\!\cdots\!35\)\( T^{33} + \)\(51\!\cdots\!66\)\( T^{34} + \)\(12\!\cdots\!34\)\( T^{35} + \)\(35\!\cdots\!28\)\( T^{36} + \)\(53\!\cdots\!29\)\( T^{37} + \)\(10\!\cdots\!07\)\( T^{38} \)
$47$ \( 1 + 40 T + 1054 T^{2} + 20442 T^{3} + 330523 T^{4} + 4599591 T^{5} + 57356270 T^{6} + 649908779 T^{7} + 6822561735 T^{8} + 66872497541 T^{9} + 619557355221 T^{10} + 5455180703414 T^{11} + 46061513183161 T^{12} + 374366454288919 T^{13} + 2947923948738331 T^{14} + 22533311972141893 T^{15} + 167907295819012305 T^{16} + 1219811438775488341 T^{17} + 8661349157140592161 T^{18} + 60044945349966815104 T^{19} + \)\(40\!\cdots\!67\)\( T^{20} + \)\(26\!\cdots\!69\)\( T^{21} + \)\(17\!\cdots\!15\)\( T^{22} + \)\(10\!\cdots\!33\)\( T^{23} + \)\(67\!\cdots\!17\)\( T^{24} + \)\(40\!\cdots\!51\)\( T^{25} + \)\(23\!\cdots\!43\)\( T^{26} + \)\(12\!\cdots\!54\)\( T^{27} + \)\(69\!\cdots\!07\)\( T^{28} + \)\(35\!\cdots\!09\)\( T^{29} + \)\(16\!\cdots\!05\)\( T^{30} + \)\(75\!\cdots\!39\)\( T^{31} + \)\(31\!\cdots\!90\)\( T^{32} + \)\(11\!\cdots\!79\)\( T^{33} + \)\(39\!\cdots\!89\)\( T^{34} + \)\(11\!\cdots\!82\)\( T^{35} + \)\(28\!\cdots\!98\)\( T^{36} + \)\(50\!\cdots\!60\)\( T^{37} + \)\(58\!\cdots\!83\)\( T^{38} \)
$53$ \( 1 + 3 T + 412 T^{2} + 998 T^{3} + 88593 T^{4} + 177385 T^{5} + 13348140 T^{6} + 22498941 T^{7} + 1580929971 T^{8} + 2273397458 T^{9} + 155879855657 T^{10} + 193226190074 T^{11} + 13213544307947 T^{12} + 14274004331766 T^{13} + 982162404430577 T^{14} + 940719453256547 T^{15} + 64825708012701211 T^{16} + 56418963351739284 T^{17} + 3827617461797003727 T^{18} + 3112679846710590048 T^{19} + \)\(20\!\cdots\!31\)\( T^{20} + \)\(15\!\cdots\!56\)\( T^{21} + \)\(96\!\cdots\!47\)\( T^{22} + \)\(74\!\cdots\!07\)\( T^{23} + \)\(41\!\cdots\!61\)\( T^{24} + \)\(31\!\cdots\!14\)\( T^{25} + \)\(15\!\cdots\!39\)\( T^{26} + \)\(12\!\cdots\!14\)\( T^{27} + \)\(51\!\cdots\!81\)\( T^{28} + \)\(39\!\cdots\!42\)\( T^{29} + \)\(14\!\cdots\!87\)\( T^{30} + \)\(11\!\cdots\!81\)\( T^{31} + \)\(34\!\cdots\!20\)\( T^{32} + \)\(24\!\cdots\!65\)\( T^{33} + \)\(64\!\cdots\!01\)\( T^{34} + \)\(38\!\cdots\!58\)\( T^{35} + \)\(84\!\cdots\!56\)\( T^{36} + \)\(32\!\cdots\!67\)\( T^{37} + \)\(57\!\cdots\!17\)\( T^{38} \)
$59$ \( 1 + 28 T + 834 T^{2} + 16479 T^{3} + 311813 T^{4} + 4943073 T^{5} + 74128844 T^{6} + 1002325477 T^{7} + 12868359737 T^{8} + 153615164976 T^{9} + 1751527999988 T^{10} + 18854002002405 T^{11} + 194823653348045 T^{12} + 1916799758931661 T^{13} + 18167671021569848 T^{14} + 164792443459516498 T^{15} + 1443189299739530571 T^{16} + 12132068388706314585 T^{17} + 98584033478702479197 T^{18} + \)\(77\!\cdots\!48\)\( T^{19} + \)\(58\!\cdots\!23\)\( T^{20} + \)\(42\!\cdots\!85\)\( T^{21} + \)\(29\!\cdots\!09\)\( T^{22} + \)\(19\!\cdots\!78\)\( T^{23} + \)\(12\!\cdots\!52\)\( T^{24} + \)\(80\!\cdots\!01\)\( T^{25} + \)\(48\!\cdots\!55\)\( T^{26} + \)\(27\!\cdots\!05\)\( T^{27} + \)\(15\!\cdots\!32\)\( T^{28} + \)\(78\!\cdots\!76\)\( T^{29} + \)\(38\!\cdots\!83\)\( T^{30} + \)\(17\!\cdots\!37\)\( T^{31} + \)\(77\!\cdots\!76\)\( T^{32} + \)\(30\!\cdots\!53\)\( T^{33} + \)\(11\!\cdots\!87\)\( T^{34} + \)\(35\!\cdots\!39\)\( T^{35} + \)\(10\!\cdots\!46\)\( T^{36} + \)\(21\!\cdots\!88\)\( T^{37} + \)\(44\!\cdots\!39\)\( T^{38} \)
$61$ \( 1 + 46 T + 1446 T^{2} + 33221 T^{3} + 645076 T^{4} + 10758279 T^{5} + 162008665 T^{6} + 2216232209 T^{7} + 28251487284 T^{8} + 336372180939 T^{9} + 3798545625234 T^{10} + 40686985743326 T^{11} + 417543179461438 T^{12} + 4100061559925217 T^{13} + 38797876798529420 T^{14} + 353056458691331983 T^{15} + 3106272555322987178 T^{16} + 26354879086383658111 T^{17} + \)\(21\!\cdots\!62\)\( T^{18} + \)\(17\!\cdots\!34\)\( T^{19} + \)\(13\!\cdots\!82\)\( T^{20} + \)\(98\!\cdots\!31\)\( T^{21} + \)\(70\!\cdots\!18\)\( T^{22} + \)\(48\!\cdots\!03\)\( T^{23} + \)\(32\!\cdots\!20\)\( T^{24} + \)\(21\!\cdots\!37\)\( T^{25} + \)\(13\!\cdots\!98\)\( T^{26} + \)\(77\!\cdots\!06\)\( T^{27} + \)\(44\!\cdots\!94\)\( T^{28} + \)\(23\!\cdots\!39\)\( T^{29} + \)\(12\!\cdots\!24\)\( T^{30} + \)\(58\!\cdots\!89\)\( T^{31} + \)\(26\!\cdots\!65\)\( T^{32} + \)\(10\!\cdots\!39\)\( T^{33} + \)\(38\!\cdots\!76\)\( T^{34} + \)\(12\!\cdots\!81\)\( T^{35} + \)\(32\!\cdots\!66\)\( T^{36} + \)\(62\!\cdots\!26\)\( T^{37} + \)\(83\!\cdots\!41\)\( T^{38} \)
$67$ \( 1 + 42 T + 1575 T^{2} + 39878 T^{3} + 908009 T^{4} + 16969283 T^{5} + 290907710 T^{6} + 4373018796 T^{7} + 61331474684 T^{8} + 780283200836 T^{9} + 9405518849175 T^{10} + 105231969621198 T^{11} + 1131917940357961 T^{12} + 11498704416367879 T^{13} + 113622583192346603 T^{14} + 1071932815093372898 T^{15} + 9891341460679541474 T^{16} + 87446819829371019123 T^{17} + \)\(75\!\cdots\!50\)\( T^{18} + \)\(62\!\cdots\!98\)\( T^{19} + \)\(50\!\cdots\!50\)\( T^{20} + \)\(39\!\cdots\!47\)\( T^{21} + \)\(29\!\cdots\!62\)\( T^{22} + \)\(21\!\cdots\!58\)\( T^{23} + \)\(15\!\cdots\!21\)\( T^{24} + \)\(10\!\cdots\!51\)\( T^{25} + \)\(68\!\cdots\!03\)\( T^{26} + \)\(42\!\cdots\!18\)\( T^{27} + \)\(25\!\cdots\!25\)\( T^{28} + \)\(14\!\cdots\!64\)\( T^{29} + \)\(74\!\cdots\!72\)\( T^{30} + \)\(35\!\cdots\!56\)\( T^{31} + \)\(15\!\cdots\!70\)\( T^{32} + \)\(62\!\cdots\!07\)\( T^{33} + \)\(22\!\cdots\!87\)\( T^{34} + \)\(65\!\cdots\!18\)\( T^{35} + \)\(17\!\cdots\!25\)\( T^{36} + \)\(31\!\cdots\!78\)\( T^{37} + \)\(49\!\cdots\!03\)\( T^{38} \)
$71$ \( 1 + 46 T + 1416 T^{2} + 31979 T^{3} + 610138 T^{4} + 10061445 T^{5} + 150970465 T^{6} + 2079366941 T^{7} + 26960668746 T^{8} + 329623790121 T^{9} + 3852796195242 T^{10} + 43000970977110 T^{11} + 462356794054962 T^{12} + 4780075558168587 T^{13} + 47827348859556946 T^{14} + 462049404462359699 T^{15} + 4331617893151475920 T^{16} + 39302095847057353817 T^{17} + \)\(34\!\cdots\!18\)\( T^{18} + \)\(29\!\cdots\!46\)\( T^{19} + \)\(24\!\cdots\!78\)\( T^{20} + \)\(19\!\cdots\!97\)\( T^{21} + \)\(15\!\cdots\!20\)\( T^{22} + \)\(11\!\cdots\!19\)\( T^{23} + \)\(86\!\cdots\!46\)\( T^{24} + \)\(61\!\cdots\!27\)\( T^{25} + \)\(42\!\cdots\!42\)\( T^{26} + \)\(27\!\cdots\!10\)\( T^{27} + \)\(17\!\cdots\!02\)\( T^{28} + \)\(10\!\cdots\!21\)\( T^{29} + \)\(62\!\cdots\!66\)\( T^{30} + \)\(34\!\cdots\!81\)\( T^{31} + \)\(17\!\cdots\!15\)\( T^{32} + \)\(83\!\cdots\!45\)\( T^{33} + \)\(35\!\cdots\!38\)\( T^{34} + \)\(13\!\cdots\!59\)\( T^{35} + \)\(41\!\cdots\!56\)\( T^{36} + \)\(96\!\cdots\!06\)\( T^{37} + \)\(14\!\cdots\!31\)\( T^{38} \)
$73$ \( 1 - 31 T + 1294 T^{2} - 29370 T^{3} + 741362 T^{4} - 13655959 T^{5} + 263513554 T^{6} - 4142758475 T^{7} + 66520605294 T^{8} - 918746991765 T^{9} + 12799746997835 T^{10} - 158132703251184 T^{11} + 1957544533024645 T^{12} - 21886220625034447 T^{13} + 244276817435380915 T^{14} - 2490133454428234836 T^{15} + 25287937970432991546 T^{16} - \)\(23\!\cdots\!44\)\( T^{17} + \)\(21\!\cdots\!99\)\( T^{18} - \)\(18\!\cdots\!84\)\( T^{19} + \)\(16\!\cdots\!27\)\( T^{20} - \)\(12\!\cdots\!76\)\( T^{21} + \)\(98\!\cdots\!82\)\( T^{22} - \)\(70\!\cdots\!76\)\( T^{23} + \)\(50\!\cdots\!95\)\( T^{24} - \)\(33\!\cdots\!83\)\( T^{25} + \)\(21\!\cdots\!65\)\( T^{26} - \)\(12\!\cdots\!04\)\( T^{27} + \)\(75\!\cdots\!55\)\( T^{28} - \)\(39\!\cdots\!85\)\( T^{29} + \)\(20\!\cdots\!38\)\( T^{30} - \)\(94\!\cdots\!75\)\( T^{31} + \)\(44\!\cdots\!82\)\( T^{32} - \)\(16\!\cdots\!31\)\( T^{33} + \)\(66\!\cdots\!34\)\( T^{34} - \)\(19\!\cdots\!70\)\( T^{35} + \)\(61\!\cdots\!82\)\( T^{36} - \)\(10\!\cdots\!39\)\( T^{37} + \)\(25\!\cdots\!37\)\( T^{38} \)
$79$ \( 1 + 56 T + 1751 T^{2} + 40251 T^{3} + 768165 T^{4} + 12982463 T^{5} + 200498578 T^{6} + 2871717245 T^{7} + 38535946843 T^{8} + 488856506209 T^{9} + 5906021798132 T^{10} + 68290067433246 T^{11} + 758626958932316 T^{12} + 8125309517850771 T^{13} + 84172064243536311 T^{14} + 845374015350145971 T^{15} + 8245992766585914749 T^{16} + 78232615790179343445 T^{17} + \)\(72\!\cdots\!46\)\( T^{18} + \)\(65\!\cdots\!74\)\( T^{19} + \)\(57\!\cdots\!34\)\( T^{20} + \)\(48\!\cdots\!45\)\( T^{21} + \)\(40\!\cdots\!11\)\( T^{22} + \)\(32\!\cdots\!51\)\( T^{23} + \)\(25\!\cdots\!89\)\( T^{24} + \)\(19\!\cdots\!91\)\( T^{25} + \)\(14\!\cdots\!44\)\( T^{26} + \)\(10\!\cdots\!06\)\( T^{27} + \)\(70\!\cdots\!08\)\( T^{28} + \)\(46\!\cdots\!09\)\( T^{29} + \)\(28\!\cdots\!97\)\( T^{30} + \)\(16\!\cdots\!45\)\( T^{31} + \)\(93\!\cdots\!42\)\( T^{32} + \)\(47\!\cdots\!03\)\( T^{33} + \)\(22\!\cdots\!35\)\( T^{34} + \)\(92\!\cdots\!71\)\( T^{35} + \)\(31\!\cdots\!09\)\( T^{36} + \)\(80\!\cdots\!16\)\( T^{37} + \)\(11\!\cdots\!19\)\( T^{38} \)
$83$ \( 1 + 25 T + 1051 T^{2} + 20585 T^{3} + 510946 T^{4} + 8368347 T^{5} + 157295265 T^{6} + 2243056623 T^{7} + 35101450564 T^{8} + 448135896138 T^{9} + 6135824348909 T^{10} + 71553214836029 T^{11} + 883311764321072 T^{12} + 9542482089027602 T^{13} + 108253626347253710 T^{14} + 1093294901011008236 T^{15} + 11535043282104115171 T^{16} + \)\(10\!\cdots\!89\)\( T^{17} + \)\(10\!\cdots\!07\)\( T^{18} + \)\(96\!\cdots\!72\)\( T^{19} + \)\(89\!\cdots\!81\)\( T^{20} + \)\(75\!\cdots\!21\)\( T^{21} + \)\(65\!\cdots\!77\)\( T^{22} + \)\(51\!\cdots\!56\)\( T^{23} + \)\(42\!\cdots\!30\)\( T^{24} + \)\(31\!\cdots\!38\)\( T^{25} + \)\(23\!\cdots\!44\)\( T^{26} + \)\(16\!\cdots\!89\)\( T^{27} + \)\(11\!\cdots\!27\)\( T^{28} + \)\(69\!\cdots\!62\)\( T^{29} + \)\(45\!\cdots\!88\)\( T^{30} + \)\(23\!\cdots\!03\)\( T^{31} + \)\(13\!\cdots\!95\)\( T^{32} + \)\(61\!\cdots\!63\)\( T^{33} + \)\(31\!\cdots\!22\)\( T^{34} + \)\(10\!\cdots\!85\)\( T^{35} + \)\(44\!\cdots\!73\)\( T^{36} + \)\(87\!\cdots\!25\)\( T^{37} + \)\(29\!\cdots\!47\)\( T^{38} \)
$89$ \( 1 + 7 T + 964 T^{2} + 4454 T^{3} + 437542 T^{4} + 1166643 T^{5} + 128078168 T^{6} + 140924113 T^{7} + 27834247843 T^{8} - 1650657100 T^{9} + 4860448158536 T^{10} - 3964771674614 T^{11} + 714056828966148 T^{12} - 879984127345524 T^{13} + 90678035997531389 T^{14} - 128074711932031809 T^{15} + 10126341138806063407 T^{16} - 14697176129571376618 T^{17} + \)\(10\!\cdots\!90\)\( T^{18} - \)\(14\!\cdots\!40\)\( T^{19} + \)\(89\!\cdots\!10\)\( T^{20} - \)\(11\!\cdots\!78\)\( T^{21} + \)\(71\!\cdots\!83\)\( T^{22} - \)\(80\!\cdots\!69\)\( T^{23} + \)\(50\!\cdots\!61\)\( T^{24} - \)\(43\!\cdots\!64\)\( T^{25} + \)\(31\!\cdots\!92\)\( T^{26} - \)\(15\!\cdots\!34\)\( T^{27} + \)\(17\!\cdots\!24\)\( T^{28} - \)\(51\!\cdots\!00\)\( T^{29} + \)\(77\!\cdots\!27\)\( T^{30} + \)\(34\!\cdots\!73\)\( T^{31} + \)\(28\!\cdots\!92\)\( T^{32} + \)\(22\!\cdots\!63\)\( T^{33} + \)\(76\!\cdots\!58\)\( T^{34} + \)\(69\!\cdots\!94\)\( T^{35} + \)\(13\!\cdots\!56\)\( T^{36} + \)\(85\!\cdots\!67\)\( T^{37} + \)\(10\!\cdots\!09\)\( T^{38} \)
$97$ \( 1 - 39 T + 1812 T^{2} - 48290 T^{3} + 1343996 T^{4} - 28015607 T^{5} + 591847844 T^{6} - 10331773501 T^{7} + 181372882043 T^{8} - 2762894411788 T^{9} + 42260825462822 T^{10} - 576732763030202 T^{11} + 7902903390374710 T^{12} - 98237879336580364 T^{13} + 1226269306796116981 T^{14} - 14025839064140651091 T^{15} + \)\(16\!\cdots\!39\)\( T^{16} - \)\(17\!\cdots\!58\)\( T^{17} + \)\(18\!\cdots\!48\)\( T^{18} - \)\(17\!\cdots\!32\)\( T^{19} + \)\(17\!\cdots\!56\)\( T^{20} - \)\(16\!\cdots\!22\)\( T^{21} + \)\(14\!\cdots\!47\)\( T^{22} - \)\(12\!\cdots\!71\)\( T^{23} + \)\(10\!\cdots\!17\)\( T^{24} - \)\(81\!\cdots\!56\)\( T^{25} + \)\(63\!\cdots\!30\)\( T^{26} - \)\(45\!\cdots\!22\)\( T^{27} + \)\(32\!\cdots\!74\)\( T^{28} - \)\(20\!\cdots\!12\)\( T^{29} + \)\(12\!\cdots\!79\)\( T^{30} - \)\(71\!\cdots\!41\)\( T^{31} + \)\(39\!\cdots\!88\)\( T^{32} - \)\(18\!\cdots\!83\)\( T^{33} + \)\(85\!\cdots\!28\)\( T^{34} - \)\(29\!\cdots\!90\)\( T^{35} + \)\(10\!\cdots\!44\)\( T^{36} - \)\(22\!\cdots\!71\)\( T^{37} + \)\(56\!\cdots\!33\)\( T^{38} \)
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