Properties

Label 8001.2.a.v
Level 8001
Weight 2
Character orbit 8001.a
Self dual yes
Analytic conductor 63.888
Analytic rank 0
Dimension 19
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2667)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{10} q^{5} + q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{10} q^{5} + q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} + ( \beta_{1} - \beta_{6} - \beta_{12} + \beta_{17} ) q^{10} + \beta_{8} q^{11} + ( 1 + \beta_{12} ) q^{13} -\beta_{1} q^{14} + ( \beta_{2} + \beta_{8} + \beta_{9} ) q^{16} + ( -1 + \beta_{7} - \beta_{16} ) q^{17} + ( 2 - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{11} - \beta_{16} ) q^{19} + ( -\beta_{5} + \beta_{10} + \beta_{12} ) q^{20} + ( -\beta_{1} - \beta_{3} - \beta_{7} + \beta_{15} ) q^{22} + ( 1 + \beta_{2} - \beta_{6} + \beta_{10} + \beta_{18} ) q^{23} + ( 3 + \beta_{2} - \beta_{9} - \beta_{15} - \beta_{16} + \beta_{18} ) q^{25} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - 2 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{16} - \beta_{18} ) q^{26} + ( 1 + \beta_{2} ) q^{28} + ( 1 - \beta_{1} - \beta_{7} + \beta_{10} + \beta_{11} + \beta_{18} ) q^{29} + ( 1 + \beta_{10} - \beta_{13} - \beta_{17} ) q^{31} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{14} + \beta_{15} ) q^{32} + ( 1 + \beta_{1} - \beta_{5} - \beta_{10} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{18} ) q^{34} + \beta_{10} q^{35} + ( 3 - \beta_{6} + \beta_{10} + \beta_{14} + \beta_{18} ) q^{37} + ( -1 - \beta_{1} + \beta_{5} - 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{18} ) q^{38} + ( -2 + \beta_{2} + \beta_{4} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{13} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{40} + ( 1 + \beta_{1} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{41} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{10} + \beta_{12} + \beta_{16} - \beta_{17} ) q^{43} + ( 1 + 3 \beta_{2} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{13} - 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{44} + ( 1 - 2 \beta_{1} - \beta_{6} + 2 \beta_{10} + \beta_{13} + \beta_{16} ) q^{46} + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{10} + \beta_{13} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{47} + q^{49} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{50} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{15} + \beta_{17} ) q^{52} + ( -2 + \beta_{1} + \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{15} - \beta_{16} - \beta_{18} ) q^{53} + ( 2 + \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{17} + \beta_{18} ) q^{55} + ( -\beta_{1} - \beta_{3} ) q^{56} + ( \beta_{1} + \beta_{5} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{13} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{58} + ( -1 - \beta_{1} + \beta_{3} - \beta_{6} + 2 \beta_{10} + 2 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} + \beta_{18} ) q^{59} + ( 1 - \beta_{1} + \beta_{5} - \beta_{7} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{16} ) q^{61} + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{13} + \beta_{16} + 2 \beta_{17} ) q^{62} + ( 4 - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{64} + ( -1 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} + 4 \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{65} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{67} + ( -\beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{68} + ( \beta_{1} - \beta_{6} - \beta_{12} + \beta_{17} ) q^{70} + ( 2 + \beta_{2} + \beta_{3} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{16} - \beta_{18} ) q^{71} + ( 3 + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{9} - \beta_{11} - \beta_{15} + \beta_{17} ) q^{73} + ( -2 \beta_{1} + \beta_{3} - \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{13} - \beta_{14} + \beta_{16} ) q^{74} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{15} + \beta_{16} ) q^{76} + \beta_{8} q^{77} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{15} - \beta_{17} ) q^{79} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{10} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{17} ) q^{80} + ( -2 - 2 \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{82} + ( \beta_{2} - \beta_{3} + \beta_{6} - \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{83} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{16} - \beta_{17} ) q^{85} + ( 2 \beta_{2} + \beta_{9} - 3 \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{86} + ( -1 - 6 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{11} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} ) q^{88} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{14} - \beta_{15} - 2 \beta_{16} + \beta_{17} - \beta_{18} ) q^{89} + ( 1 + \beta_{12} ) q^{91} + ( 4 - \beta_{1} + 3 \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} ) q^{92} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{17} + \beta_{18} ) q^{94} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{95} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{97} -\beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19q - 4q^{2} + 22q^{4} - 5q^{5} + 19q^{7} - 9q^{8} + O(q^{10}) \) \( 19q - 4q^{2} + 22q^{4} - 5q^{5} + 19q^{7} - 9q^{8} + 9q^{11} + 24q^{13} - 4q^{14} + 20q^{16} - 17q^{17} + 23q^{19} - 5q^{20} - 3q^{22} + 17q^{23} + 38q^{25} - 28q^{26} + 22q^{28} - 2q^{29} + 16q^{31} - 17q^{32} + 29q^{34} - 5q^{35} + 56q^{37} - 2q^{38} - 13q^{40} + 7q^{41} + 19q^{43} + 29q^{44} + 10q^{46} - 25q^{47} + 19q^{49} + 9q^{50} + 16q^{52} - 18q^{53} + 10q^{55} - 9q^{56} + 31q^{58} - 11q^{59} + 26q^{61} - 26q^{62} + 45q^{64} - 27q^{65} + 24q^{67} - 14q^{68} + 32q^{71} + 51q^{73} + 12q^{76} + 9q^{77} + 30q^{79} + 30q^{80} - 52q^{82} - q^{83} + 44q^{85} + 24q^{86} - 30q^{88} - 5q^{89} + 24q^{91} + 88q^{92} + 7q^{94} + 24q^{95} + 5q^{97} - 4q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + 21 x^{11} - 17032 x^{10} + 4985 x^{9} + 29792 x^{8} - 13249 x^{7} - 28600 x^{6} + 14000 x^{5} + 13725 x^{4} - 5723 x^{3} - 2913 x^{2} + 608 x + 210\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\(11967908 \nu^{18} + 10330919 \nu^{17} - 422890858 \nu^{16} - 185012986 \nu^{15} + 5929350010 \nu^{14} + 1055361561 \nu^{13} - 43092462154 \nu^{12} - 1159028944 \nu^{11} + 177057585547 \nu^{10} - 8859733933 \nu^{9} - 419706704682 \nu^{8} + 32463907935 \nu^{7} + 560761757568 \nu^{6} - 42523921020 \nu^{5} - 394138444469 \nu^{4} + 26352010178 \nu^{3} + 124055335687 \nu^{2} - 9280440934 \nu - 9272820218\)\()/ 579103703 \)
\(\beta_{5}\)\(=\)\((\)\(65114179 \nu^{18} - 132926940 \nu^{17} - 1671831523 \nu^{16} + 3285269092 \nu^{15} + 17368062950 \nu^{14} - 32561720490 \nu^{13} - 94107501027 \nu^{12} + 166375659285 \nu^{11} + 285664374904 \nu^{10} - 469100157707 \nu^{9} - 486402019943 \nu^{8} + 726626199106 \nu^{7} + 448072525313 \nu^{6} - 582478309828 \nu^{5} - 222447224597 \nu^{4} + 206901778527 \nu^{3} + 71510619076 \nu^{2} - 23752461453 \nu - 6128233325\)\()/ 2895518515 \)
\(\beta_{6}\)\(=\)\((\)\(138645363 \nu^{18} - 343698145 \nu^{17} - 3866824621 \nu^{16} + 9112645479 \nu^{15} + 44954282930 \nu^{14} - 98469596705 \nu^{13} - 284325273789 \nu^{12} + 559933954855 \nu^{11} + 1071360077368 \nu^{10} - 1803665054044 \nu^{9} - 2478737837371 \nu^{8} + 3295314382092 \nu^{7} + 3500770767616 \nu^{6} - 3225526895581 \nu^{5} - 2872131807524 \nu^{4} + 1473343414389 \nu^{3} + 1168583011302 \nu^{2} - 242700042466 \nu - 133199717500\)\()/ 5791037030 \)
\(\beta_{7}\)\(=\)\((\)\(318716503 \nu^{18} - 1140866135 \nu^{17} - 7501449651 \nu^{16} + 29542139284 \nu^{15} + 67595392300 \nu^{14} - 309929331465 \nu^{13} - 281749529274 \nu^{12} + 1696786234950 \nu^{11} + 451103298313 \nu^{10} - 5196857369249 \nu^{9} + 375217402964 \nu^{8} + 8851478759162 \nu^{7} - 2162801145399 \nu^{6} - 7808662348931 \nu^{5} + 2393312892281 \nu^{4} + 2961548762254 \nu^{3} - 839166774003 \nu^{2} - 274887004146 \nu + 54702171430\)\()/ 11582074060 \)
\(\beta_{8}\)\(=\)\((\)\(-93609108 \nu^{18} + 195011795 \nu^{17} + 2566480776 \nu^{16} - 4911478344 \nu^{15} - 29294218685 \nu^{14} + 49897035585 \nu^{13} + 181459391204 \nu^{12} - 263787283760 \nu^{11} - 664790856488 \nu^{10} + 781891378959 \nu^{9} + 1466329373136 \nu^{8} - 1310268250057 \nu^{7} - 1887181446716 \nu^{6} + 1200884819526 \nu^{5} + 1292110192984 \nu^{4} - 547670340074 \nu^{3} - 388056127817 \nu^{2} + 88500571266 \nu + 40044597930\)\()/ 2895518515 \)
\(\beta_{9}\)\(=\)\((\)\(93609108 \nu^{18} - 195011795 \nu^{17} - 2566480776 \nu^{16} + 4911478344 \nu^{15} + 29294218685 \nu^{14} - 49897035585 \nu^{13} - 181459391204 \nu^{12} + 263787283760 \nu^{11} + 664790856488 \nu^{10} - 781891378959 \nu^{9} - 1466329373136 \nu^{8} + 1310268250057 \nu^{7} + 1887181446716 \nu^{6} - 1200884819526 \nu^{5} - 1289214674469 \nu^{4} + 547670340074 \nu^{3} + 367787498212 \nu^{2} - 88500571266 \nu - 19775968325\)\()/ 2895518515 \)
\(\beta_{10}\)\(=\)\((\)\(-37953488 \nu^{18} + 92196111 \nu^{17} + 991696752 \nu^{16} - 2324375148 \nu^{15} - 10672704577 \nu^{14} + 23745817195 \nu^{13} + 61534319327 \nu^{12} - 126913768361 \nu^{11} - 206178478291 \nu^{10} + 381112300142 \nu^{9} + 404304603883 \nu^{8} - 638066586796 \nu^{7} - 439470286205 \nu^{6} + 548407227881 \nu^{5} + 230702575789 \nu^{4} - 194336671842 \nu^{3} - 44204269770 \nu^{2} + 15157612018 \nu + 2068831656\)\()/ 1158207406 \)
\(\beta_{11}\)\(=\)\((\)\(-240105167 \nu^{18} + 881338715 \nu^{17} + 5495659804 \nu^{16} - 21651992771 \nu^{15} - 49786456290 \nu^{14} + 215066182570 \nu^{13} + 231300589556 \nu^{12} - 1119608505200 \nu^{11} - 605499617407 \nu^{10} + 3307955295931 \nu^{9} + 961255585349 \nu^{8} - 5591713835683 \nu^{7} - 1060121848224 \nu^{6} + 5135218235504 \nu^{5} + 906752365551 \nu^{4} - 2201652718176 \nu^{3} - 455951257588 \nu^{2} + 287009377064 \nu + 53652831630\)\()/ 5791037030 \)
\(\beta_{12}\)\(=\)\((\)\(290773938 \nu^{18} - 662122355 \nu^{17} - 7697023186 \nu^{16} + 16815829469 \nu^{15} + 83341546495 \nu^{14} - 172281371545 \nu^{13} - 477184016914 \nu^{12} + 916160179920 \nu^{11} + 1554464292428 \nu^{10} - 2705691492119 \nu^{9} - 2868279746356 \nu^{8} + 4389554678512 \nu^{7} + 2773105119646 \nu^{6} - 3584941413081 \nu^{5} - 1121219155374 \nu^{4} + 1145884621839 \nu^{3} + 63474192337 \nu^{2} - 53779704136 \nu + 19309791560\)\()/ 5791037030 \)
\(\beta_{13}\)\(=\)\((\)\(-485961931 \nu^{18} + 1113420900 \nu^{17} + 13007283232 \nu^{16} - 28691667118 \nu^{15} - 142770006155 \nu^{14} + 298637498085 \nu^{13} + 833859292088 \nu^{12} - 1614947153800 \nu^{11} - 2816533502951 \nu^{10} + 4855571864208 \nu^{9} + 5615791269077 \nu^{8} - 8046688000049 \nu^{7} - 6493467022012 \nu^{6} + 6795149697847 \nu^{5} + 4085986343623 \nu^{4} - 2351933693773 \nu^{3} - 1176825672339 \nu^{2} + 161926194492 \nu + 74147034150\)\()/ 5791037030 \)
\(\beta_{14}\)\(=\)\((\)\(-244538816 \nu^{18} + 640007340 \nu^{17} + 6214873087 \nu^{16} - 15917066553 \nu^{15} - 64356454145 \nu^{14} + 159447001730 \nu^{13} + 351910167043 \nu^{12} - 829200724505 \nu^{11} - 1098123323401 \nu^{10} + 2402070934223 \nu^{9} + 1964936315422 \nu^{8} - 3854034717714 \nu^{7} - 1924408194587 \nu^{6} + 3182220496297 \nu^{5} + 959561891823 \nu^{4} - 1107518683308 \nu^{3} - 255693379414 \nu^{2} + 88860693382 \nu + 28681664520\)\()/ 2895518515 \)
\(\beta_{15}\)\(=\)\((\)\(1036415051 \nu^{18} - 3169187735 \nu^{17} - 25673615907 \nu^{16} + 80069329128 \nu^{15} + 255548957080 \nu^{14} - 817470456425 \nu^{13} - 1312960193338 \nu^{12} + 4348086495830 \nu^{11} + 3700939092301 \nu^{10} - 12928740475313 \nu^{9} - 5538919778952 \nu^{8} + 21361112833594 \nu^{7} + 3742541531697 \nu^{6} - 18219213168867 \nu^{5} - 555145776623 \nu^{4} + 6668255047918 \nu^{3} - 102435732651 \nu^{2} - 709051042762 \nu - 23929479290\)\()/ 11582074060 \)
\(\beta_{16}\)\(=\)\((\)\(-1177420721 \nu^{18} + 3022806135 \nu^{17} + 30214864317 \nu^{16} - 76041669698 \nu^{15} - 316167323730 \nu^{14} + 773094156925 \nu^{13} + 1747199273478 \nu^{12} - 4098812389930 \nu^{11} - 5505446366291 \nu^{10} + 12181293389813 \nu^{9} + 9922760972072 \nu^{8} - 20239239560654 \nu^{7} - 9698265109967 \nu^{6} + 17589452904587 \nu^{5} + 4616300955633 \nu^{4} - 6763494721768 \nu^{3} - 1017780451669 \nu^{2} + 804092581722 \nu + 101431035930\)\()/ 11582074060 \)
\(\beta_{17}\)\(=\)\((\)\(727508506 \nu^{18} - 1789420580 \nu^{17} - 19108483507 \nu^{16} + 45513393513 \nu^{15} + 205976043850 \nu^{14} - 467788147365 \nu^{13} - 1184324624578 \nu^{12} + 2503001409990 \nu^{11} + 3952381907166 \nu^{10} - 7476870253978 \nu^{9} - 7810236222227 \nu^{8} + 12396449304189 \nu^{7} + 8959188531857 \nu^{6} - 10620725347607 \nu^{5} - 5626225717688 \nu^{4} + 3926288444198 \nu^{3} + 1709061696269 \nu^{2} - 427993545432 \nu - 153741088340\)\()/ 5791037030 \)
\(\beta_{18}\)\(=\)\((\)\(3366745827 \nu^{18} - 9263333215 \nu^{17} - 85049828069 \nu^{16} + 231373969696 \nu^{15} + 877139208820 \nu^{14} - 2334094800655 \nu^{13} - 4806101920096 \nu^{12} + 12274426272960 \nu^{11} + 15248263236747 \nu^{10} - 36169011220831 \nu^{9} - 28613582641044 \nu^{8} + 59507326593548 \nu^{7} + 31177087450109 \nu^{6} - 50962697977249 \nu^{5} - 18711356928001 \nu^{4} + 18937119806716 \nu^{3} + 5559535596393 \nu^{2} - 1938290594534 \nu - 509625256950\)\()/ 11582074060 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{9} + \beta_{8} + 7 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(-\beta_{15} - \beta_{14} + \beta_{7} - \beta_{5} + 9 \beta_{3} + 30 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(\beta_{17} - \beta_{16} - 2 \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{11} + 11 \beta_{9} + 11 \beta_{8} + \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + 46 \beta_{2} + 80\)
\(\nu^{7}\)\(=\)\(\beta_{18} + \beta_{17} - \beta_{16} - 14 \beta_{15} - 12 \beta_{14} + 2 \beta_{13} + 2 \beta_{11} + 3 \beta_{10} + \beta_{9} + \beta_{8} + 14 \beta_{7} - 14 \beta_{5} + 70 \beta_{3} - \beta_{2} + 195 \beta_{1} + 14\)
\(\nu^{8}\)\(=\)\(2 \beta_{18} + 13 \beta_{17} - 14 \beta_{16} - 31 \beta_{15} - 13 \beta_{14} + 16 \beta_{13} + 4 \beta_{12} + 31 \beta_{11} + 4 \beta_{10} + 96 \beta_{9} + 95 \beta_{8} + 17 \beta_{7} + 30 \beta_{6} - \beta_{5} - 29 \beta_{4} + 2 \beta_{3} + 305 \beta_{2} - 2 \beta_{1} + 508\)
\(\nu^{9}\)\(=\)\(17 \beta_{18} + 17 \beta_{17} - 16 \beta_{16} - 145 \beta_{15} - 109 \beta_{14} + 34 \beta_{13} + 8 \beta_{12} + 32 \beta_{11} + 54 \beta_{10} + 16 \beta_{9} + 20 \beta_{8} + 142 \beta_{7} + 3 \beta_{6} - 143 \beta_{5} - \beta_{4} + 525 \beta_{3} - 12 \beta_{2} + 1321 \beta_{1} + 141\)
\(\nu^{10}\)\(=\)\(38 \beta_{18} + 123 \beta_{17} - 137 \beta_{16} - 338 \beta_{15} - 125 \beta_{14} + 181 \beta_{13} + 78 \beta_{12} + 334 \beta_{11} + 80 \beta_{10} + 777 \beta_{9} + 762 \beta_{8} + 199 \beta_{7} + 318 \beta_{6} - 21 \beta_{5} - 295 \beta_{4} + 39 \beta_{3} + 2061 \beta_{2} - 28 \beta_{1} + 3420\)
\(\nu^{11}\)\(=\)\(196 \beta_{18} + 201 \beta_{17} - 175 \beta_{16} - 1338 \beta_{15} - 902 \beta_{14} + 401 \beta_{13} + 165 \beta_{12} + 354 \beta_{11} + 651 \beta_{10} + 185 \beta_{9} + 258 \beta_{8} + 1281 \beta_{7} + 63 \beta_{6} - 1291 \beta_{5} - 22 \beta_{4} + 3896 \beta_{3} - 92 \beta_{2} + 9180 \beta_{1} + 1270\)
\(\nu^{12}\)\(=\)\(472 \beta_{18} + 1052 \beta_{17} - 1168 \beta_{16} - 3209 \beta_{15} - 1087 \beta_{14} + 1783 \beta_{13} + 1001 \beta_{12} + 3109 \beta_{11} + 1036 \beta_{10} + 6089 \beta_{9} + 5939 \beta_{8} + 1997 \beta_{7} + 2936 \beta_{6} - 291 \beta_{5} - 2617 \beta_{4} + 509 \beta_{3} + 14191 \beta_{2} - 222 \beta_{1} + 23845\)
\(\nu^{13}\)\(=\)\(1930 \beta_{18} + 2048 \beta_{17} - 1628 \beta_{16} - 11637 \beta_{15} - 7176 \beta_{14} + 4070 \beta_{13} + 2196 \beta_{12} + 3394 \beta_{11} + 6618 \beta_{10} + 1887 \beta_{9} + 2764 \beta_{8} + 10931 \beta_{7} + 856 \beta_{6} - 10946 \beta_{5} - 312 \beta_{4} + 28856 \beta_{3} - 508 \beta_{2} + 64948 \beta_{1} + 10927\)
\(\nu^{14}\)\(=\)\(4884 \beta_{18} + 8685 \beta_{17} - 9323 \beta_{16} - 28414 \beta_{15} - 9063 \beta_{14} + 16327 \beta_{13} + 10700 \beta_{12} + 26856 \beta_{11} + 11110 \beta_{10} + 46978 \beta_{9} + 45726 \beta_{8} + 18471 \beta_{7} + 25254 \beta_{6} - 3357 \beta_{5} - 21718 \beta_{4} + 5607 \beta_{3} + 99346 \beta_{2} - 954 \beta_{1} + 170076\)
\(\nu^{15}\)\(=\)\(17519 \beta_{18} + 19249 \beta_{17} - 13913 \beta_{16} - 97749 \beta_{15} - 56041 \beta_{14} + 38110 \beta_{13} + 24088 \beta_{12} + 30340 \beta_{11} + 61369 \beta_{10} + 18027 \beta_{9} + 26842 \beta_{8} + 90467 \beta_{7} + 9628 \beta_{6} - 89511 \beta_{5} - 3656 \beta_{4} + 214022 \beta_{3} - 1303 \beta_{2} + 465784 \beta_{1} + 92083\)
\(\nu^{16}\)\(=\)\(45824 \beta_{18} + 70814 \beta_{17} - 71931 \beta_{16} - 241742 \beta_{15} - 74068 \beta_{14} + 142969 \beta_{13} + 103396 \beta_{12} + 222271 \beta_{11} + 107566 \beta_{10} + 359574 \beta_{9} + 350215 \beta_{8} + 162701 \beta_{7} + 208480 \beta_{6} - 34917 \beta_{5} - 173795 \beta_{4} + 56387 \beta_{3} + 705414 \beta_{2} + 4037 \beta_{1} + 1232416\)
\(\nu^{17}\)\(=\)\(151648 \beta_{18} + 171997 \beta_{17} - 113356 \beta_{16} - 803134 \beta_{15} - 433642 \beta_{14} + 339563 \beta_{13} + 237420 \beta_{12} + 260928 \beta_{11} + 538331 \beta_{10} + 165372 \beta_{9} + 245930 \beta_{8} + 734926 \beta_{7} + 97683 \beta_{6} - 715742 \beta_{5} - 38593 \beta_{4} + 1591650 \beta_{3} + 15758 \beta_{2} + 3376342 \beta_{1} + 767873\)
\(\nu^{18}\)\(=\)\(405545 \beta_{18} + 574450 \beta_{17} - 545365 \beta_{16} - 2006325 \beta_{15} - 599014 \beta_{14} + 1215030 \beta_{13} + 938630 \beta_{12} + 1792211 \beta_{11} + 978901 \beta_{10} + 2741034 \beta_{9} + 2676791 \beta_{8} + 1388627 \beta_{7} + 1677626 \beta_{6} - 339940 \beta_{5} - 1361630 \beta_{4} + 536340 \beta_{3} + 5069187 \beta_{2} + 153210 \beta_{1} + 9034683\)

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
2.79341
2.49323
2.42782
2.26573
1.88777
1.49024
1.35771
1.20823
0.823573
0.395540
−0.249163
−0.407951
−0.782325
−1.30029
−1.48888
−1.91759
−1.94177
−2.35941
−2.69590
−2.79341 0 5.80316 1.63704 0 1.00000 −10.6238 0 −4.57293
1.2 −2.49323 0 4.21621 0.262973 0 1.00000 −5.52554 0 −0.655653
1.3 −2.42782 0 3.89430 −4.13136 0 1.00000 −4.59902 0 10.0302
1.4 −2.26573 0 3.13354 0.263451 0 1.00000 −2.56830 0 −0.596909
1.5 −1.88777 0 1.56369 3.95777 0 1.00000 0.823654 0 −7.47138
1.6 −1.49024 0 0.220813 −2.04744 0 1.00000 2.65141 0 3.05118
1.7 −1.35771 0 −0.156613 −4.06757 0 1.00000 2.92806 0 5.52260
1.8 −1.20823 0 −0.540168 −0.486834 0 1.00000 3.06912 0 0.588209
1.9 −0.823573 0 −1.32173 2.87784 0 1.00000 2.73569 0 −2.37011
1.10 −0.395540 0 −1.84355 −1.95228 0 1.00000 1.52028 0 0.772204
1.11 0.249163 0 −1.93792 −0.989651 0 1.00000 −0.981182 0 −0.246584
1.12 0.407951 0 −1.83358 1.08992 0 1.00000 −1.56391 0 0.444635
1.13 0.782325 0 −1.38797 2.85652 0 1.00000 −2.65049 0 2.23472
1.14 1.30029 0 −0.309249 −3.89453 0 1.00000 −3.00269 0 −5.06401
1.15 1.48888 0 0.216778 2.91159 0 1.00000 −2.65501 0 4.33502
1.16 1.91759 0 1.67713 −1.63006 0 1.00000 −0.619124 0 −3.12578
1.17 1.94177 0 1.77048 −3.66642 0 1.00000 −0.445683 0 −7.11935
1.18 2.35941 0 3.56679 3.48367 0 1.00000 3.69670 0 8.21939
1.19 2.69590 0 5.26787 −1.47463 0 1.00000 8.80983 0 −3.97544
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

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 8001.2.a.v 19
3.b odd 2 1 2667.2.a.q 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.q 19 3.b odd 2 1
8001.2.a.v 19 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{19} + \cdots\)
\(T_{5}^{19} + \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 16 T^{2} + 43 T^{3} + 114 T^{4} + 251 T^{5} + 541 T^{6} + 1040 T^{7} + 1959 T^{8} + 3404 T^{9} + 5807 T^{10} + 9344 T^{11} + 14813 T^{12} + 22536 T^{13} + 33898 T^{14} + 49667 T^{15} + 72273 T^{16} + 103641 T^{17} + 148038 T^{18} + 209658 T^{19} + 296076 T^{20} + 414564 T^{21} + 578184 T^{22} + 794672 T^{23} + 1084736 T^{24} + 1442304 T^{25} + 1896064 T^{26} + 2392064 T^{27} + 2973184 T^{28} + 3485696 T^{29} + 4012032 T^{30} + 4259840 T^{31} + 4431872 T^{32} + 4112384 T^{33} + 3735552 T^{34} + 2818048 T^{35} + 2097152 T^{36} + 1048576 T^{37} + 524288 T^{38} \)
$3$ \( \)
$5$ \( 1 + 5 T + 41 T^{2} + 168 T^{3} + 832 T^{4} + 2996 T^{5} + 11691 T^{6} + 37986 T^{7} + 128552 T^{8} + 382033 T^{9} + 1168939 T^{10} + 3222334 T^{11} + 9102821 T^{12} + 23575655 T^{13} + 62285480 T^{14} + 152712478 T^{15} + 380888120 T^{16} + 886961463 T^{17} + 2101604027 T^{18} + 4656991668 T^{19} + 10508020135 T^{20} + 22174036575 T^{21} + 47611015000 T^{22} + 95445298750 T^{23} + 194642125000 T^{24} + 368369609375 T^{25} + 711157890625 T^{26} + 1258724218750 T^{27} + 2283083984375 T^{28} + 3730791015625 T^{29} + 6276953125000 T^{30} + 9273925781250 T^{31} + 14271240234375 T^{32} + 18286132812500 T^{33} + 25390625000000 T^{34} + 25634765625000 T^{35} + 31280517578125 T^{36} + 19073486328125 T^{37} + 19073486328125 T^{38} \)
$7$ \( ( 1 - T )^{19} \)
$11$ \( 1 - 9 T + 114 T^{2} - 846 T^{3} + 6834 T^{4} - 43631 T^{5} + 279049 T^{6} - 1570491 T^{7} + 8622246 T^{8} - 43583626 T^{9} + 213037611 T^{10} - 979860254 T^{11} + 4350940113 T^{12} - 18383133454 T^{13} + 75002259850 T^{14} - 292955898389 T^{15} + 1105867682544 T^{16} - 4009634203784 T^{17} + 14057469809074 T^{18} - 47407746375208 T^{19} + 154632167899814 T^{20} - 485165738657864 T^{21} + 1471909885466064 T^{22} - 4289167308313349 T^{23} + 12079188951102350 T^{24} - 32566842284901694 T^{25} + 84787513992790323 T^{26} - 210041747583815774 T^{27} + 502331542953606201 T^{28} - 1130447013213183226 T^{29} + 2460027410679012306 T^{30} - 4928873519784940011 T^{31} + 9633528301051801619 T^{32} - 16568864989070388071 T^{33} + 28547313989786558934 T^{34} - 38873471464582048206 T^{35} + 57620961248919489894 T^{36} - 50039255821430083329 T^{37} + 61159090448414546291 T^{38} \)
$13$ \( 1 - 24 T + 402 T^{2} - 5026 T^{3} + 52972 T^{4} - 482098 T^{5} + 3925963 T^{6} - 28966932 T^{7} + 196682920 T^{8} - 1238002338 T^{9} + 7283995209 T^{10} - 40243236364 T^{11} + 209809038801 T^{12} - 1035189853758 T^{13} + 4848463373224 T^{14} - 21594807026076 T^{15} + 91636684117992 T^{16} - 370834674951958 T^{17} + 1432622313733372 T^{18} - 5285005449943492 T^{19} + 18624090078533836 T^{20} - 62671060066880902 T^{21} + 201325795007228424 T^{22} - 616769283471756636 T^{23} + 1800200511234458632 T^{24} - 4996663702827798222 T^{25} + 13165206037958208117 T^{26} - 32827644214579138444 T^{27} + 77243122626775503957 T^{28} - \)\(17\!\cdots\!62\)\( T^{29} + \)\(35\!\cdots\!40\)\( T^{30} - \)\(67\!\cdots\!92\)\( T^{31} + \)\(11\!\cdots\!39\)\( T^{32} - \)\(18\!\cdots\!22\)\( T^{33} + \)\(27\!\cdots\!04\)\( T^{34} - \)\(33\!\cdots\!66\)\( T^{35} + \)\(34\!\cdots\!66\)\( T^{36} - \)\(26\!\cdots\!96\)\( T^{37} + \)\(14\!\cdots\!77\)\( T^{38} \)
$17$ \( 1 + 17 T + 284 T^{2} + 3184 T^{3} + 32551 T^{4} + 278609 T^{5} + 2179544 T^{6} + 15312493 T^{7} + 99697602 T^{8} + 601566144 T^{9} + 3415708339 T^{10} + 18303102870 T^{11} + 93623696982 T^{12} + 457896515162 T^{13} + 2163836903127 T^{14} + 9881182119343 T^{15} + 43988887312946 T^{16} + 190801647107236 T^{17} + 811083332879856 T^{18} + 3375158517221308 T^{19} + 13788416658957552 T^{20} + 55141676013991204 T^{21} + 216117403368503698 T^{22} + 825286211789646703 T^{23} + 3072338973763192839 T^{24} + 11052508729582321178 T^{25} + 38417423580947984886 T^{26} + \)\(12\!\cdots\!70\)\( T^{27} + \)\(40\!\cdots\!83\)\( T^{28} + \)\(12\!\cdots\!56\)\( T^{29} + \)\(34\!\cdots\!66\)\( T^{30} + \)\(89\!\cdots\!73\)\( T^{31} + \)\(21\!\cdots\!28\)\( T^{32} + \)\(46\!\cdots\!61\)\( T^{33} + \)\(93\!\cdots\!43\)\( T^{34} + \)\(15\!\cdots\!04\)\( T^{35} + \)\(23\!\cdots\!68\)\( T^{36} + \)\(23\!\cdots\!53\)\( T^{37} + \)\(23\!\cdots\!53\)\( T^{38} \)
$19$ \( 1 - 23 T + 416 T^{2} - 5333 T^{3} + 59693 T^{4} - 564360 T^{5} + 4868647 T^{6} - 37688990 T^{7} + 272710055 T^{8} - 1823898623 T^{9} + 11572107859 T^{10} - 69036155352 T^{11} + 394624910977 T^{12} - 2144671314045 T^{13} + 11249450770817 T^{14} - 56533025766972 T^{15} + 275667627613884 T^{16} - 1294507948679753 T^{17} + 5918690016197407 T^{18} - 26130047561244090 T^{19} + 112455110307750733 T^{20} - 467317369473390833 T^{21} + 1890804257803630356 T^{22} - 7367440450977558012 T^{23} + 27854753804169202883 T^{24} - \)\(10\!\cdots\!45\)\( T^{25} + \)\(35\!\cdots\!03\)\( T^{26} - \)\(11\!\cdots\!32\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} - \)\(11\!\cdots\!23\)\( T^{29} + \)\(31\!\cdots\!45\)\( T^{30} - \)\(83\!\cdots\!90\)\( T^{31} + \)\(20\!\cdots\!73\)\( T^{32} - \)\(45\!\cdots\!60\)\( T^{33} + \)\(90\!\cdots\!07\)\( T^{34} - \)\(15\!\cdots\!73\)\( T^{35} + \)\(22\!\cdots\!24\)\( T^{36} - \)\(23\!\cdots\!43\)\( T^{37} + \)\(19\!\cdots\!79\)\( T^{38} \)
$23$ \( 1 - 17 T + 391 T^{2} - 4769 T^{3} + 65448 T^{4} - 640527 T^{5} + 6681855 T^{6} - 55642304 T^{7} + 483186514 T^{8} - 3544873790 T^{9} + 26828147273 T^{10} - 177193952328 T^{11} + 1199885605017 T^{12} - 7232532287266 T^{13} + 44528435308294 T^{14} - 247034068307600 T^{15} + 1396581507639034 T^{16} - 7165658273752720 T^{17} + 37417693376472169 T^{18} - 177916417007843598 T^{19} + 860606947658859887 T^{20} - 3790633226815188880 T^{21} + 16992207203444126678 T^{22} - 69130260709267091600 T^{23} + \)\(28\!\cdots\!42\)\( T^{24} - \)\(10\!\cdots\!74\)\( T^{25} + \)\(40\!\cdots\!99\)\( T^{26} - \)\(13\!\cdots\!68\)\( T^{27} + \)\(48\!\cdots\!99\)\( T^{28} - \)\(14\!\cdots\!10\)\( T^{29} + \)\(46\!\cdots\!78\)\( T^{30} - \)\(12\!\cdots\!84\)\( T^{31} + \)\(33\!\cdots\!65\)\( T^{32} - \)\(74\!\cdots\!43\)\( T^{33} + \)\(17\!\cdots\!36\)\( T^{34} - \)\(29\!\cdots\!09\)\( T^{35} + \)\(55\!\cdots\!73\)\( T^{36} - \)\(55\!\cdots\!73\)\( T^{37} + \)\(74\!\cdots\!87\)\( T^{38} \)
$29$ \( 1 + 2 T + 250 T^{2} + 416 T^{3} + 31743 T^{4} + 40300 T^{5} + 2734376 T^{6} + 2380316 T^{7} + 180736146 T^{8} + 93979076 T^{9} + 9822415154 T^{10} + 2467083804 T^{11} + 457248535436 T^{12} + 31843829608 T^{13} + 18656669806136 T^{14} - 735202286540 T^{15} + 675508971482301 T^{16} - 63722501247098 T^{17} + 21852698540725853 T^{18} - 2340535489419288 T^{19} + 633728257681049737 T^{20} - 53590623548809418 T^{21} + 16474988305481839089 T^{22} - 519994608426297740 T^{23} + \)\(38\!\cdots\!64\)\( T^{24} + 18941452480788688168 T^{25} + \)\(78\!\cdots\!24\)\( T^{26} + \)\(12\!\cdots\!44\)\( T^{27} + \)\(14\!\cdots\!26\)\( T^{28} + \)\(39\!\cdots\!76\)\( T^{29} + \)\(22\!\cdots\!34\)\( T^{30} + \)\(84\!\cdots\!56\)\( T^{31} + \)\(28\!\cdots\!64\)\( T^{32} + \)\(11\!\cdots\!00\)\( T^{33} + \)\(27\!\cdots\!07\)\( T^{34} + \)\(10\!\cdots\!36\)\( T^{35} + \)\(18\!\cdots\!50\)\( T^{36} + \)\(42\!\cdots\!22\)\( T^{37} + \)\(61\!\cdots\!69\)\( T^{38} \)
$31$ \( 1 - 16 T + 444 T^{2} - 5891 T^{3} + 94707 T^{4} - 1068715 T^{5} + 12949771 T^{6} - 127334359 T^{7} + 1280574803 T^{8} - 11191113269 T^{9} + 97892201385 T^{10} - 771667588986 T^{11} + 6026996383307 T^{12} - 43307805557421 T^{13} + 306799204066761 T^{14} - 2023856746847967 T^{15} + 13129737685672700 T^{16} - 79857016141786523 T^{17} + 477117795202829813 T^{18} - 2680713867518439722 T^{19} + 14790651651287724203 T^{20} - 76742592512256848603 T^{21} + \)\(39\!\cdots\!00\)\( T^{22} - \)\(18\!\cdots\!07\)\( T^{23} + \)\(87\!\cdots\!11\)\( T^{24} - \)\(38\!\cdots\!01\)\( T^{25} + \)\(16\!\cdots\!77\)\( T^{26} - \)\(65\!\cdots\!26\)\( T^{27} + \)\(25\!\cdots\!35\)\( T^{28} - \)\(91\!\cdots\!69\)\( T^{29} + \)\(32\!\cdots\!93\)\( T^{30} - \)\(10\!\cdots\!99\)\( T^{31} + \)\(31\!\cdots\!61\)\( T^{32} - \)\(80\!\cdots\!15\)\( T^{33} + \)\(22\!\cdots\!57\)\( T^{34} - \)\(42\!\cdots\!71\)\( T^{35} + \)\(10\!\cdots\!84\)\( T^{36} - \)\(11\!\cdots\!56\)\( T^{37} + \)\(21\!\cdots\!71\)\( T^{38} \)
$37$ \( 1 - 56 T + 1919 T^{2} - 48113 T^{3} + 978205 T^{4} - 16860046 T^{5} + 254586879 T^{6} - 3434890242 T^{7} + 42066352834 T^{8} - 472720007895 T^{9} + 4918349057024 T^{10} - 47690421274916 T^{11} + 433348998146773 T^{12} - 3705349253519629 T^{13} + 29916574318889081 T^{14} - 228662300563992558 T^{15} + 1658031228676445269 T^{16} - 11421713005692758978 T^{17} + 74832242387766162423 T^{18} - \)\(46\!\cdots\!78\)\( T^{19} + \)\(27\!\cdots\!51\)\( T^{20} - \)\(15\!\cdots\!82\)\( T^{21} + \)\(83\!\cdots\!57\)\( T^{22} - \)\(42\!\cdots\!38\)\( T^{23} + \)\(20\!\cdots\!17\)\( T^{24} - \)\(95\!\cdots\!61\)\( T^{25} + \)\(41\!\cdots\!09\)\( T^{26} - \)\(16\!\cdots\!36\)\( T^{27} + \)\(63\!\cdots\!48\)\( T^{28} - \)\(22\!\cdots\!55\)\( T^{29} + \)\(74\!\cdots\!42\)\( T^{30} - \)\(22\!\cdots\!02\)\( T^{31} + \)\(62\!\cdots\!63\)\( T^{32} - \)\(15\!\cdots\!94\)\( T^{33} + \)\(32\!\cdots\!65\)\( T^{34} - \)\(59\!\cdots\!33\)\( T^{35} + \)\(87\!\cdots\!23\)\( T^{36} - \)\(94\!\cdots\!24\)\( T^{37} + \)\(62\!\cdots\!73\)\( T^{38} \)
$41$ \( 1 - 7 T + 372 T^{2} - 2494 T^{3} + 72192 T^{4} - 464718 T^{5} + 9659139 T^{6} - 59703885 T^{7} + 994893253 T^{8} - 5895564041 T^{9} + 83515099021 T^{10} - 473396571593 T^{11} + 5907087984900 T^{12} - 31939933276623 T^{13} + 359380353673970 T^{14} - 1847303925885007 T^{15} + 19050148050782657 T^{16} - 92701601847103523 T^{17} + 886536429602488599 T^{18} - 4062791660052265530 T^{19} + 36347993613702032559 T^{20} - \)\(15\!\cdots\!63\)\( T^{21} + \)\(13\!\cdots\!97\)\( T^{22} - \)\(52\!\cdots\!27\)\( T^{23} + \)\(41\!\cdots\!70\)\( T^{24} - \)\(15\!\cdots\!43\)\( T^{25} + \)\(11\!\cdots\!00\)\( T^{26} - \)\(37\!\cdots\!53\)\( T^{27} + \)\(27\!\cdots\!81\)\( T^{28} - \)\(79\!\cdots\!41\)\( T^{29} + \)\(54\!\cdots\!73\)\( T^{30} - \)\(13\!\cdots\!85\)\( T^{31} + \)\(89\!\cdots\!19\)\( T^{32} - \)\(17\!\cdots\!98\)\( T^{33} + \)\(11\!\cdots\!92\)\( T^{34} - \)\(15\!\cdots\!54\)\( T^{35} + \)\(97\!\cdots\!32\)\( T^{36} - \)\(75\!\cdots\!47\)\( T^{37} + \)\(43\!\cdots\!61\)\( T^{38} \)
$43$ \( 1 - 19 T + 541 T^{2} - 8177 T^{3} + 138902 T^{4} - 1756335 T^{5} + 22810067 T^{6} - 249749424 T^{7} + 2714308406 T^{8} - 26376020952 T^{9} + 251013493425 T^{10} - 2205434613904 T^{11} + 18891895734477 T^{12} - 152330616533600 T^{13} + 1197167503618210 T^{14} - 8968364455463072 T^{15} + 65582001060951154 T^{16} - 460827672059565390 T^{17} + 3166245237871279593 T^{18} - 20989222701232991902 T^{19} + \)\(13\!\cdots\!99\)\( T^{20} - \)\(85\!\cdots\!10\)\( T^{21} + \)\(52\!\cdots\!78\)\( T^{22} - \)\(30\!\cdots\!72\)\( T^{23} + \)\(17\!\cdots\!30\)\( T^{24} - \)\(96\!\cdots\!00\)\( T^{25} + \)\(51\!\cdots\!39\)\( T^{26} - \)\(25\!\cdots\!04\)\( T^{27} + \)\(12\!\cdots\!75\)\( T^{28} - \)\(57\!\cdots\!48\)\( T^{29} + \)\(25\!\cdots\!42\)\( T^{30} - \)\(99\!\cdots\!24\)\( T^{31} + \)\(39\!\cdots\!81\)\( T^{32} - \)\(12\!\cdots\!15\)\( T^{33} + \)\(44\!\cdots\!14\)\( T^{34} - \)\(11\!\cdots\!77\)\( T^{35} + \)\(31\!\cdots\!63\)\( T^{36} - \)\(47\!\cdots\!31\)\( T^{37} + \)\(10\!\cdots\!07\)\( T^{38} \)
$47$ \( 1 + 25 T + 904 T^{2} + 16894 T^{3} + 361649 T^{4} + 5490414 T^{5} + 88619801 T^{6} + 1143888312 T^{7} + 15210985459 T^{8} + 171652595794 T^{9} + 1965291833203 T^{10} + 19743875550578 T^{11} + 199637602810875 T^{12} + 1807202412082554 T^{13} + 16393786057089957 T^{14} + 134786478179220970 T^{15} + 1107872880178406762 T^{16} + 8312551665909679693 T^{17} + 62279110124776561297 T^{18} + \)\(42\!\cdots\!32\)\( T^{19} + \)\(29\!\cdots\!59\)\( T^{20} + \)\(18\!\cdots\!37\)\( T^{21} + \)\(11\!\cdots\!26\)\( T^{22} + \)\(65\!\cdots\!70\)\( T^{23} + \)\(37\!\cdots\!99\)\( T^{24} + \)\(19\!\cdots\!66\)\( T^{25} + \)\(10\!\cdots\!25\)\( T^{26} + \)\(47\!\cdots\!58\)\( T^{27} + \)\(21\!\cdots\!01\)\( T^{28} + \)\(90\!\cdots\!06\)\( T^{29} + \)\(37\!\cdots\!77\)\( T^{30} + \)\(13\!\cdots\!92\)\( T^{31} + \)\(48\!\cdots\!27\)\( T^{32} + \)\(14\!\cdots\!66\)\( T^{33} + \)\(43\!\cdots\!07\)\( T^{34} + \)\(95\!\cdots\!74\)\( T^{35} + \)\(24\!\cdots\!48\)\( T^{36} + \)\(31\!\cdots\!25\)\( T^{37} + \)\(58\!\cdots\!83\)\( T^{38} \)
$53$ \( 1 + 18 T + 729 T^{2} + 10786 T^{3} + 248005 T^{4} + 3156553 T^{5} + 53632959 T^{6} + 604817569 T^{7} + 8403382051 T^{8} + 85661456697 T^{9} + 1025414824379 T^{10} + 9578389113218 T^{11} + 101873882940499 T^{12} + 879848025198707 T^{13} + 8477677851613975 T^{14} + 68061668751127647 T^{15} + 601679391573782590 T^{16} + 4501409880480265273 T^{17} + 36809960564576726564 T^{18} + \)\(25\!\cdots\!92\)\( T^{19} + \)\(19\!\cdots\!92\)\( T^{20} + \)\(12\!\cdots\!57\)\( T^{21} + \)\(89\!\cdots\!30\)\( T^{22} + \)\(53\!\cdots\!07\)\( T^{23} + \)\(35\!\cdots\!75\)\( T^{24} + \)\(19\!\cdots\!03\)\( T^{25} + \)\(11\!\cdots\!63\)\( T^{26} + \)\(59\!\cdots\!98\)\( T^{27} + \)\(33\!\cdots\!07\)\( T^{28} + \)\(14\!\cdots\!53\)\( T^{29} + \)\(77\!\cdots\!47\)\( T^{30} + \)\(29\!\cdots\!29\)\( T^{31} + \)\(13\!\cdots\!07\)\( T^{32} + \)\(43\!\cdots\!57\)\( T^{33} + \)\(18\!\cdots\!85\)\( T^{34} + \)\(41\!\cdots\!06\)\( T^{35} + \)\(14\!\cdots\!77\)\( T^{36} + \)\(19\!\cdots\!02\)\( T^{37} + \)\(57\!\cdots\!17\)\( T^{38} \)
$59$ \( 1 + 11 T + 625 T^{2} + 6723 T^{3} + 195828 T^{4} + 2071013 T^{5} + 41248883 T^{6} + 425750946 T^{7} + 6571435112 T^{8} + 65378594338 T^{9} + 841495712425 T^{10} + 7973031579512 T^{11} + 89681549643081 T^{12} + 802152520437678 T^{13} + 8123426402651024 T^{14} + 68246189918775774 T^{15} + 633654102053130070 T^{16} + 4987112049848041672 T^{17} + 42904033250390068767 T^{18} + \)\(31\!\cdots\!46\)\( T^{19} + \)\(25\!\cdots\!53\)\( T^{20} + \)\(17\!\cdots\!32\)\( T^{21} + \)\(13\!\cdots\!30\)\( T^{22} + \)\(82\!\cdots\!14\)\( T^{23} + \)\(58\!\cdots\!76\)\( T^{24} + \)\(33\!\cdots\!98\)\( T^{25} + \)\(22\!\cdots\!39\)\( T^{26} + \)\(11\!\cdots\!52\)\( T^{27} + \)\(72\!\cdots\!75\)\( T^{28} + \)\(33\!\cdots\!38\)\( T^{29} + \)\(19\!\cdots\!08\)\( T^{30} + \)\(75\!\cdots\!26\)\( T^{31} + \)\(43\!\cdots\!57\)\( T^{32} + \)\(12\!\cdots\!93\)\( T^{33} + \)\(71\!\cdots\!72\)\( T^{34} + \)\(14\!\cdots\!43\)\( T^{35} + \)\(79\!\cdots\!75\)\( T^{36} + \)\(82\!\cdots\!31\)\( T^{37} + \)\(44\!\cdots\!39\)\( T^{38} \)
$61$ \( 1 - 26 T + 838 T^{2} - 15329 T^{3} + 298583 T^{4} - 4294165 T^{5} + 63615119 T^{6} - 761644687 T^{7} + 9351456495 T^{8} - 96581854497 T^{9} + 1031428248461 T^{10} - 9443877771964 T^{11} + 90986491118717 T^{12} - 758071858895611 T^{13} + 6812650245376485 T^{14} - 53082743698333913 T^{15} + 458408958260406348 T^{16} - 3427992803436643349 T^{17} + 29015749356968724265 T^{18} - \)\(21\!\cdots\!10\)\( T^{19} + \)\(17\!\cdots\!65\)\( T^{20} - \)\(12\!\cdots\!29\)\( T^{21} + \)\(10\!\cdots\!88\)\( T^{22} - \)\(73\!\cdots\!33\)\( T^{23} + \)\(57\!\cdots\!85\)\( T^{24} - \)\(39\!\cdots\!71\)\( T^{25} + \)\(28\!\cdots\!57\)\( T^{26} - \)\(18\!\cdots\!84\)\( T^{27} + \)\(12\!\cdots\!01\)\( T^{28} - \)\(68\!\cdots\!97\)\( T^{29} + \)\(40\!\cdots\!95\)\( T^{30} - \)\(20\!\cdots\!27\)\( T^{31} + \)\(10\!\cdots\!39\)\( T^{32} - \)\(42\!\cdots\!65\)\( T^{33} + \)\(17\!\cdots\!83\)\( T^{34} - \)\(56\!\cdots\!69\)\( T^{35} + \)\(18\!\cdots\!98\)\( T^{36} - \)\(35\!\cdots\!06\)\( T^{37} + \)\(83\!\cdots\!41\)\( T^{38} \)
$67$ \( 1 - 24 T + 1065 T^{2} - 20598 T^{3} + 528809 T^{4} - 8644973 T^{5} + 165769208 T^{6} - 2361709360 T^{7} + 37198542309 T^{8} - 471412532131 T^{9} + 6396164360286 T^{10} - 73119929927616 T^{11} + 878356408508454 T^{12} - 9144679664174609 T^{13} + 98919019711057509 T^{14} - 943722960976640048 T^{15} + 9291719328933613139 T^{16} - 81516068920894131031 T^{17} + \)\(73\!\cdots\!96\)\( T^{18} - \)\(59\!\cdots\!16\)\( T^{19} + \)\(49\!\cdots\!32\)\( T^{20} - \)\(36\!\cdots\!59\)\( T^{21} + \)\(27\!\cdots\!57\)\( T^{22} - \)\(19\!\cdots\!08\)\( T^{23} + \)\(13\!\cdots\!63\)\( T^{24} - \)\(82\!\cdots\!21\)\( T^{25} + \)\(53\!\cdots\!42\)\( T^{26} - \)\(29\!\cdots\!56\)\( T^{27} + \)\(17\!\cdots\!42\)\( T^{28} - \)\(85\!\cdots\!19\)\( T^{29} + \)\(45\!\cdots\!47\)\( T^{30} - \)\(19\!\cdots\!60\)\( T^{31} + \)\(90\!\cdots\!96\)\( T^{32} - \)\(31\!\cdots\!17\)\( T^{33} + \)\(13\!\cdots\!87\)\( T^{34} - \)\(33\!\cdots\!38\)\( T^{35} + \)\(11\!\cdots\!55\)\( T^{36} - \)\(17\!\cdots\!16\)\( T^{37} + \)\(49\!\cdots\!03\)\( T^{38} \)
$71$ \( 1 - 32 T + 1227 T^{2} - 28835 T^{3} + 680427 T^{4} - 12842100 T^{5} + 234406211 T^{6} - 3740306330 T^{7} + 57238298625 T^{8} - 796894059719 T^{9} + 10634242395334 T^{10} - 131843183828886 T^{11} + 1569149036104108 T^{12} - 17561592195883763 T^{13} + 189022343080292513 T^{14} - 1927041274911330686 T^{15} + 18922276766488552162 T^{16} - \)\(17\!\cdots\!86\)\( T^{17} + \)\(15\!\cdots\!38\)\( T^{18} - \)\(13\!\cdots\!34\)\( T^{19} + \)\(11\!\cdots\!98\)\( T^{20} - \)\(89\!\cdots\!26\)\( T^{21} + \)\(67\!\cdots\!82\)\( T^{22} - \)\(48\!\cdots\!66\)\( T^{23} + \)\(34\!\cdots\!63\)\( T^{24} - \)\(22\!\cdots\!23\)\( T^{25} + \)\(14\!\cdots\!28\)\( T^{26} - \)\(85\!\cdots\!46\)\( T^{27} + \)\(48\!\cdots\!54\)\( T^{28} - \)\(25\!\cdots\!19\)\( T^{29} + \)\(13\!\cdots\!75\)\( T^{30} - \)\(61\!\cdots\!30\)\( T^{31} + \)\(27\!\cdots\!21\)\( T^{32} - \)\(10\!\cdots\!00\)\( T^{33} + \)\(39\!\cdots\!77\)\( T^{34} - \)\(12\!\cdots\!35\)\( T^{35} + \)\(36\!\cdots\!57\)\( T^{36} - \)\(67\!\cdots\!52\)\( T^{37} + \)\(14\!\cdots\!31\)\( T^{38} \)
$73$ \( 1 - 51 T + 2082 T^{2} - 60320 T^{3} + 1520884 T^{4} - 32377455 T^{5} + 624053883 T^{6} - 10764805837 T^{7} + 171927543584 T^{8} - 2527267835368 T^{9} + 34877971935503 T^{10} - 450100748228450 T^{11} + 5502830968614309 T^{12} - 63523695181462276 T^{13} + 698837312980657430 T^{14} - 7303189643834057099 T^{15} + 73007994854064934486 T^{16} - \)\(69\!\cdots\!98\)\( T^{17} + \)\(63\!\cdots\!58\)\( T^{18} - \)\(55\!\cdots\!84\)\( T^{19} + \)\(46\!\cdots\!34\)\( T^{20} - \)\(37\!\cdots\!42\)\( T^{21} + \)\(28\!\cdots\!62\)\( T^{22} - \)\(20\!\cdots\!59\)\( T^{23} + \)\(14\!\cdots\!90\)\( T^{24} - \)\(96\!\cdots\!64\)\( T^{25} + \)\(60\!\cdots\!73\)\( T^{26} - \)\(36\!\cdots\!50\)\( T^{27} + \)\(20\!\cdots\!39\)\( T^{28} - \)\(10\!\cdots\!32\)\( T^{29} + \)\(53\!\cdots\!68\)\( T^{30} - \)\(24\!\cdots\!77\)\( T^{31} + \)\(10\!\cdots\!39\)\( T^{32} - \)\(39\!\cdots\!95\)\( T^{33} + \)\(13\!\cdots\!88\)\( T^{34} - \)\(39\!\cdots\!20\)\( T^{35} + \)\(98\!\cdots\!46\)\( T^{36} - \)\(17\!\cdots\!19\)\( T^{37} + \)\(25\!\cdots\!37\)\( T^{38} \)
$79$ \( 1 - 30 T + 1323 T^{2} - 30260 T^{3} + 793055 T^{4} - 14916571 T^{5} + 296531429 T^{6} - 4783765086 T^{7} + 78863642538 T^{8} - 1120203367049 T^{9} + 16007575831074 T^{10} - 203725377221417 T^{11} + 2588519002910803 T^{12} - 29871844374454987 T^{13} + 342857430889264467 T^{14} - 3616829807433295478 T^{15} + 37877700456738271513 T^{16} - \)\(36\!\cdots\!51\)\( T^{17} + \)\(35\!\cdots\!65\)\( T^{18} - \)\(31\!\cdots\!06\)\( T^{19} + \)\(27\!\cdots\!35\)\( T^{20} - \)\(22\!\cdots\!91\)\( T^{21} + \)\(18\!\cdots\!07\)\( T^{22} - \)\(14\!\cdots\!18\)\( T^{23} + \)\(10\!\cdots\!33\)\( T^{24} - \)\(72\!\cdots\!27\)\( T^{25} + \)\(49\!\cdots\!77\)\( T^{26} - \)\(30\!\cdots\!37\)\( T^{27} + \)\(19\!\cdots\!06\)\( T^{28} - \)\(10\!\cdots\!49\)\( T^{29} + \)\(58\!\cdots\!02\)\( T^{30} - \)\(28\!\cdots\!26\)\( T^{31} + \)\(13\!\cdots\!31\)\( T^{32} - \)\(55\!\cdots\!51\)\( T^{33} + \)\(23\!\cdots\!45\)\( T^{34} - \)\(69\!\cdots\!60\)\( T^{35} + \)\(24\!\cdots\!57\)\( T^{36} - \)\(43\!\cdots\!30\)\( T^{37} + \)\(11\!\cdots\!19\)\( T^{38} \)
$83$ \( 1 + T + 913 T^{2} + 180 T^{3} + 412053 T^{4} - 180103 T^{5} + 122830140 T^{6} - 112698348 T^{7} + 27225502887 T^{8} - 34293478900 T^{9} + 4787033795008 T^{10} - 7085502677768 T^{11} + 695564964961644 T^{12} - 1111539579594292 T^{13} + 85892305345946283 T^{14} - 140345510451137812 T^{15} + 9192190781252108163 T^{16} - 14778538914349895882 T^{17} + \)\(86\!\cdots\!08\)\( T^{18} - \)\(13\!\cdots\!36\)\( T^{19} + \)\(71\!\cdots\!64\)\( T^{20} - \)\(10\!\cdots\!98\)\( T^{21} + \)\(52\!\cdots\!81\)\( T^{22} - \)\(66\!\cdots\!52\)\( T^{23} + \)\(33\!\cdots\!69\)\( T^{24} - \)\(36\!\cdots\!48\)\( T^{25} + \)\(18\!\cdots\!88\)\( T^{26} - \)\(15\!\cdots\!88\)\( T^{27} + \)\(89\!\cdots\!24\)\( T^{28} - \)\(53\!\cdots\!00\)\( T^{29} + \)\(35\!\cdots\!29\)\( T^{30} - \)\(12\!\cdots\!28\)\( T^{31} + \)\(10\!\cdots\!20\)\( T^{32} - \)\(13\!\cdots\!87\)\( T^{33} + \)\(25\!\cdots\!71\)\( T^{34} + \)\(91\!\cdots\!80\)\( T^{35} + \)\(38\!\cdots\!99\)\( T^{36} + \)\(34\!\cdots\!09\)\( T^{37} + \)\(29\!\cdots\!47\)\( T^{38} \)
$89$ \( 1 + 5 T + 900 T^{2} + 5208 T^{3} + 408753 T^{4} + 2600183 T^{5} + 124544985 T^{6} + 845207354 T^{7} + 28548725824 T^{8} + 202656062858 T^{9} + 5238694079121 T^{10} + 38292815208174 T^{11} + 800260312469475 T^{12} + 5932265418022598 T^{13} + 104523943291455492 T^{14} + 772530166737831198 T^{15} + 11892279508870100778 T^{16} + 85898464458081369316 T^{17} + \)\(11\!\cdots\!55\)\( T^{18} + \)\(82\!\cdots\!80\)\( T^{19} + \)\(10\!\cdots\!95\)\( T^{20} + \)\(68\!\cdots\!36\)\( T^{21} + \)\(83\!\cdots\!82\)\( T^{22} + \)\(48\!\cdots\!18\)\( T^{23} + \)\(58\!\cdots\!08\)\( T^{24} + \)\(29\!\cdots\!78\)\( T^{25} + \)\(35\!\cdots\!75\)\( T^{26} + \)\(15\!\cdots\!94\)\( T^{27} + \)\(18\!\cdots\!89\)\( T^{28} + \)\(63\!\cdots\!58\)\( T^{29} + \)\(79\!\cdots\!36\)\( T^{30} + \)\(20\!\cdots\!34\)\( T^{31} + \)\(27\!\cdots\!65\)\( T^{32} + \)\(50\!\cdots\!03\)\( T^{33} + \)\(71\!\cdots\!97\)\( T^{34} + \)\(80\!\cdots\!88\)\( T^{35} + \)\(12\!\cdots\!00\)\( T^{36} + \)\(61\!\cdots\!05\)\( T^{37} + \)\(10\!\cdots\!09\)\( T^{38} \)
$97$ \( 1 - 5 T + 1119 T^{2} - 7425 T^{3} + 626161 T^{4} - 4997793 T^{5} + 234060233 T^{6} - 2106203122 T^{7} + 65583827219 T^{8} - 634772102388 T^{9} + 14616979308761 T^{10} - 146747567731490 T^{11} + 2681647163351003 T^{12} - 27114376204648986 T^{13} + 413553226986122049 T^{14} - 4106187033289173770 T^{15} + 54307299971976751138 T^{16} - \)\(51\!\cdots\!28\)\( T^{17} + \)\(61\!\cdots\!00\)\( T^{18} - \)\(54\!\cdots\!70\)\( T^{19} + \)\(59\!\cdots\!00\)\( T^{20} - \)\(48\!\cdots\!52\)\( T^{21} + \)\(49\!\cdots\!74\)\( T^{22} - \)\(36\!\cdots\!70\)\( T^{23} + \)\(35\!\cdots\!93\)\( T^{24} - \)\(22\!\cdots\!94\)\( T^{25} + \)\(21\!\cdots\!39\)\( T^{26} - \)\(11\!\cdots\!90\)\( T^{27} + \)\(11\!\cdots\!37\)\( T^{28} - \)\(46\!\cdots\!12\)\( T^{29} + \)\(46\!\cdots\!07\)\( T^{30} - \)\(14\!\cdots\!02\)\( T^{31} + \)\(15\!\cdots\!41\)\( T^{32} - \)\(32\!\cdots\!17\)\( T^{33} + \)\(39\!\cdots\!73\)\( T^{34} - \)\(45\!\cdots\!25\)\( T^{35} + \)\(66\!\cdots\!03\)\( T^{36} - \)\(28\!\cdots\!45\)\( T^{37} + \)\(56\!\cdots\!33\)\( T^{38} \)
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