Properties

Label 8034.2.a.bd
Level 8034
Weight 2
Character orbit 8034.a
Self dual yes
Analytic conductor 64.152
Analytic rank 0
Dimension 16
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 36 x^{14} + 196 x^{13} + 498 x^{12} - 3101 x^{11} - 3150 x^{10} + 25368 x^{9} + 6763 x^{8} - 113788 x^{7} + 19731 x^{6} + 270913 x^{5} - 122680 x^{4} - 296326 x^{3} + 185524 x^{2} + 94528 x - 66432\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + \beta_{1} q^{5} + q^{6} -\beta_{9} q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + q^{3} + q^{4} + \beta_{1} q^{5} + q^{6} -\beta_{9} q^{7} + q^{8} + q^{9} + \beta_{1} q^{10} + ( 1 + \beta_{2} ) q^{11} + q^{12} - q^{13} -\beta_{9} q^{14} + \beta_{1} q^{15} + q^{16} + ( 1 - \beta_{15} ) q^{17} + q^{18} + ( 1 - \beta_{11} ) q^{19} + \beta_{1} q^{20} -\beta_{9} q^{21} + ( 1 + \beta_{2} ) q^{22} + ( 1 + \beta_{4} ) q^{23} + q^{24} + ( 1 + \beta_{11} - \beta_{12} ) q^{25} - q^{26} + q^{27} -\beta_{9} q^{28} + ( 1 - \beta_{3} + \beta_{13} ) q^{29} + \beta_{1} q^{30} + ( \beta_{12} - \beta_{13} ) q^{31} + q^{32} + ( 1 + \beta_{2} ) q^{33} + ( 1 - \beta_{15} ) q^{34} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} + \beta_{15} ) q^{35} + q^{36} + ( 2 + \beta_{5} + \beta_{12} + \beta_{13} ) q^{37} + ( 1 - \beta_{11} ) q^{38} - q^{39} + \beta_{1} q^{40} + ( 2 + \beta_{9} - \beta_{14} ) q^{41} -\beta_{9} q^{42} + ( 2 - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{14} + \beta_{15} ) q^{43} + ( 1 + \beta_{2} ) q^{44} + \beta_{1} q^{45} + ( 1 + \beta_{4} ) q^{46} + ( 1 - \beta_{1} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{14} ) q^{47} + q^{48} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{9} - \beta_{14} ) q^{49} + ( 1 + \beta_{11} - \beta_{12} ) q^{50} + ( 1 - \beta_{15} ) q^{51} - q^{52} + ( -\beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{53} + q^{54} + ( 3 + \beta_{1} + \beta_{2} - \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{55} -\beta_{9} q^{56} + ( 1 - \beta_{11} ) q^{57} + ( 1 - \beta_{3} + \beta_{13} ) q^{58} + ( 3 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{15} ) q^{59} + \beta_{1} q^{60} + ( \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} - \beta_{15} ) q^{61} + ( \beta_{12} - \beta_{13} ) q^{62} -\beta_{9} q^{63} + q^{64} -\beta_{1} q^{65} + ( 1 + \beta_{2} ) q^{66} + ( 2 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{67} + ( 1 - \beta_{15} ) q^{68} + ( 1 + \beta_{4} ) q^{69} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} + \beta_{15} ) q^{70} + ( 1 - \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{71} + q^{72} + ( 2 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{73} + ( 2 + \beta_{5} + \beta_{12} + \beta_{13} ) q^{74} + ( 1 + \beta_{11} - \beta_{12} ) q^{75} + ( 1 - \beta_{11} ) q^{76} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{12} + \beta_{15} ) q^{77} - q^{78} + ( 1 - \beta_{1} + \beta_{2} - \beta_{7} - \beta_{8} - \beta_{10} + \beta_{12} ) q^{79} + \beta_{1} q^{80} + q^{81} + ( 2 + \beta_{9} - \beta_{14} ) q^{82} + ( 3 - \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{83} -\beta_{9} q^{84} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{85} + ( 2 - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{14} + \beta_{15} ) q^{86} + ( 1 - \beta_{3} + \beta_{13} ) q^{87} + ( 1 + \beta_{2} ) q^{88} + ( 2 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{12} ) q^{89} + \beta_{1} q^{90} + \beta_{9} q^{91} + ( 1 + \beta_{4} ) q^{92} + ( \beta_{12} - \beta_{13} ) q^{93} + ( 1 - \beta_{1} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{14} ) q^{94} + ( -1 + \beta_{2} - \beta_{3} + \beta_{8} - \beta_{11} + \beta_{12} + \beta_{15} ) q^{95} + q^{96} + ( 2 + \beta_{1} - \beta_{2} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{97} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{9} - \beta_{14} ) q^{98} + ( 1 + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{2} + 16q^{3} + 16q^{4} + 5q^{5} + 16q^{6} + 4q^{7} + 16q^{8} + 16q^{9} + O(q^{10}) \) \( 16q + 16q^{2} + 16q^{3} + 16q^{4} + 5q^{5} + 16q^{6} + 4q^{7} + 16q^{8} + 16q^{9} + 5q^{10} + 18q^{11} + 16q^{12} - 16q^{13} + 4q^{14} + 5q^{15} + 16q^{16} + 17q^{17} + 16q^{18} + 8q^{19} + 5q^{20} + 4q^{21} + 18q^{22} + 9q^{23} + 16q^{24} + 17q^{25} - 16q^{26} + 16q^{27} + 4q^{28} + 14q^{29} + 5q^{30} + 12q^{31} + 16q^{32} + 18q^{33} + 17q^{34} + 16q^{35} + 16q^{36} + 31q^{37} + 8q^{38} - 16q^{39} + 5q^{40} + 29q^{41} + 4q^{42} + 30q^{43} + 18q^{44} + 5q^{45} + 9q^{46} - q^{47} + 16q^{48} + 36q^{49} + 17q^{50} + 17q^{51} - 16q^{52} + 12q^{53} + 16q^{54} + 30q^{55} + 4q^{56} + 8q^{57} + 14q^{58} + 38q^{59} + 5q^{60} + 12q^{62} + 4q^{63} + 16q^{64} - 5q^{65} + 18q^{66} + 28q^{67} + 17q^{68} + 9q^{69} + 16q^{70} + 32q^{71} + 16q^{72} + 20q^{73} + 31q^{74} + 17q^{75} + 8q^{76} + 26q^{77} - 16q^{78} + 13q^{79} + 5q^{80} + 16q^{81} + 29q^{82} + 39q^{83} + 4q^{84} + 31q^{85} + 30q^{86} + 14q^{87} + 18q^{88} + 9q^{89} + 5q^{90} - 4q^{91} + 9q^{92} + 12q^{93} - q^{94} - 20q^{95} + 16q^{96} + 35q^{97} + 36q^{98} + 18q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 5 x^{15} - 36 x^{14} + 196 x^{13} + 498 x^{12} - 3101 x^{11} - 3150 x^{10} + 25368 x^{9} + 6763 x^{8} - 113788 x^{7} + 19731 x^{6} + 270913 x^{5} - 122680 x^{4} - 296326 x^{3} + 185524 x^{2} + 94528 x - 66432\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-90586583806696 \nu^{15} + 240231261271053 \nu^{14} + 3828384374947411 \nu^{13} - 8779176678679124 \nu^{12} - 65847084572606208 \nu^{11} + 126845466474286622 \nu^{10} + 584806300906520047 \nu^{9} - 933492931001692430 \nu^{8} - 2812402864763475044 \nu^{7} + 3770017974633780087 \nu^{6} + 7057641060272104164 \nu^{5} - 8218562721686531825 \nu^{4} - 8015723884984487975 \nu^{3} + 8434099748549824528 \nu^{2} + 2608177483732148142 \nu - 2641861718989240104\)\()/ 3920441740718852 \)
\(\beta_{3}\)\(=\)\((\)\(-548861320433565 \nu^{15} + 1432628181298625 \nu^{14} + 23250750972171732 \nu^{13} - 52222785574841876 \nu^{12} - 400715718523472458 \nu^{11} + 751279156738320473 \nu^{10} + 3564165190681044406 \nu^{9} - 5491103861096290984 \nu^{8} - 17153609961115061407 \nu^{7} + 21958491635614630188 \nu^{6} + 43039314676128770737 \nu^{5} - 47249773825946922341 \nu^{4} - 48815686138384895136 \nu^{3} + 47716865559993728766 \nu^{2} + 15822903042704832740 \nu - 14690943728166969488\)\()/ 7840883481437704 \)
\(\beta_{4}\)\(=\)\((\)\(1804326041550225 \nu^{15} - 4966726230624225 \nu^{14} - 75869762801105224 \nu^{13} + 182455633738698884 \nu^{12} + 1298746915725286210 \nu^{11} - 2653626772894102149 \nu^{10} - 11487640535802356658 \nu^{9} + 19666961745324013104 \nu^{8} + 55060744838113792651 \nu^{7} - 79839053134054770160 \nu^{6} - 137710113475680299789 \nu^{5} + 174248716157357704813 \nu^{4} + 155576241533390408220 \nu^{3} - 178441043110187005830 \nu^{2} - 49887972149323407140 \nu + 55874993881846012448\)\()/ 15681766962875408 \)
\(\beta_{5}\)\(=\)\((\)\(1835427375702341 \nu^{15} - 4835063132536829 \nu^{14} - 77588508853292304 \nu^{13} + 176229450470376708 \nu^{12} + 1334671116861721130 \nu^{11} - 2535841645574667481 \nu^{10} - 11852110311085567842 \nu^{9} + 18546698540088649616 \nu^{8} + 56965202389909391415 \nu^{7} - 74248219122217515688 \nu^{6} - 142761745839245510897 \nu^{5} + 160045111057739017785 \nu^{4} + 161726537131518925220 \nu^{3} - 162139203272019169390 \nu^{2} - 52289606909934558532 \nu + 50159255507717243952\)\()/ 15681766962875408 \)
\(\beta_{6}\)\(=\)\((\)\(-595029787973500 \nu^{15} + 1599200104156899 \nu^{14} + 25105417772097457 \nu^{13} - 58527974832066220 \nu^{12} - 431159357317343856 \nu^{11} + 846763531102328270 \nu^{10} + 3824729425828204465 \nu^{9} - 6233456803346647046 \nu^{8} - 18376977040077066764 \nu^{7} + 25116251602804037253 \nu^{6} + 46062057857375063812 \nu^{5} - 54416371328603985543 \nu^{4} - 52163888681140811825 \nu^{3} + 55327722525776352088 \nu^{2} + 16815865184121493238 \nu - 17193626618790198144\)\()/ 3920441740718852 \)
\(\beta_{7}\)\(=\)\((\)\(-2678489495580815 \nu^{15} + 7200273222605183 \nu^{14} + 113027570175626792 \nu^{13} - 263611332890321948 \nu^{12} - 1941458087986298718 \nu^{11} + 3815774167702393179 \nu^{10} + 17225700209520332286 \nu^{9} - 28110032331446166560 \nu^{8} - 82787094387047921925 \nu^{7} + 113376045704201381856 \nu^{6} + 207587078710285962883 \nu^{5} - 245983134707975915715 \nu^{4} - 235238717548900685604 \nu^{3} + 250609270874405645338 \nu^{2} + 75872710426262168508 \nu - 78040356907095709104\)\()/ 15681766962875408 \)
\(\beta_{8}\)\(=\)\((\)\(-1638703541012891 \nu^{15} + 4420432159038421 \nu^{14} + 69072367077303942 \nu^{13} - 161784188747179548 \nu^{12} - 1185014847148291254 \nu^{11} + 2341043276285811571 \nu^{10} + 10500262821630922156 \nu^{9} - 17239844280702946500 \nu^{8} - 50388167884484641945 \nu^{7} + 69503600178868582750 \nu^{6} + 126120647536918697015 \nu^{5} - 150709788164175297649 \nu^{4} - 142627782445967128474 \nu^{3} + 153429764577484876258 \nu^{2} + 45943002513497625888 \nu - 47772605674210336512\)\()/ 7840883481437704 \)
\(\beta_{9}\)\(=\)\((\)\(-3569097596475919 \nu^{15} + 9534940311240535 \nu^{14} + 150690531878100656 \nu^{13} - 348541387262623900 \nu^{12} - 2589606301822829662 \nu^{11} + 5034026218476398539 \nu^{10} + 22983893465451743718 \nu^{9} - 36974698692332377376 \nu^{8} - 110474820709431665877 \nu^{7} + 148581497019745053416 \nu^{6} + 277001407604204710851 \nu^{5} - 320980123683727999659 \nu^{4} - 313893272371210960748 \nu^{3} + 325396081581932857258 \nu^{2} + 101341860589668676780 \nu - 100871910618813386288\)\()/ 15681766962875408 \)
\(\beta_{10}\)\(=\)\((\)\(4530672367783573 \nu^{15} - 12086915535739717 \nu^{14} - 191288673559836344 \nu^{13} + 441626866537395316 \nu^{12} + 3287022861871681850 \nu^{11} - 6374316318038130041 \nu^{10} - 29168492201670204122 \nu^{9} + 46778612431374204912 \nu^{8} + 140160717342289010247 \nu^{7} - 187791522521265099936 \nu^{6} - 351310754565662542289 \nu^{5} + 405296071724458119681 \nu^{4} + 398014867364435051884 \nu^{3} - 410559868342291139342 \nu^{2} - 128593948008476186420 \nu + 127255514637495805936\)\()/ 15681766962875408 \)
\(\beta_{11}\)\(=\)\((\)\(-653898814567821 \nu^{15} + 1739613880476431 \nu^{14} + 27620987107450456 \nu^{13} - 63529149729873054 \nu^{12} - 474890744134306278 \nu^{11} + 916354503792581425 \nu^{10} + 4216832882426115422 \nu^{9} - 6719299055051450480 \nu^{8} - 20278331057027016475 \nu^{7} + 26951008155860129116 \nu^{6} + 50875551644113476007 \nu^{5} - 58121177129369561431 \nu^{4} - 57709904926601002470 \nu^{3} + 58840600922971927372 \nu^{2} + 18677578787028375660 \nu - 18225241076803877394\)\()/ 1960220870359426 \)
\(\beta_{12}\)\(=\)\((\)\(-653898814567821 \nu^{15} + 1739613880476431 \nu^{14} + 27620987107450456 \nu^{13} - 63529149729873054 \nu^{12} - 474890744134306278 \nu^{11} + 916354503792581425 \nu^{10} + 4216832882426115422 \nu^{9} - 6719299055051450480 \nu^{8} - 20278331057027016475 \nu^{7} + 26951008155860129116 \nu^{6} + 50875551644113476007 \nu^{5} - 58121177129369561431 \nu^{4} - 57709904926601002470 \nu^{3} + 58838640702101567946 \nu^{2} + 18677578787028375660 \nu - 18213479751581720838\)\()/ 1960220870359426 \)
\(\beta_{13}\)\(=\)\((\)\(3100242079114577 \nu^{15} - 8285924509613657 \nu^{14} - 130935794410224976 \nu^{13} + 303115201650576348 \nu^{12} + 2250734259315229658 \nu^{11} - 4382238950785207301 \nu^{10} - 19980925687065254106 \nu^{9} + 32228305300709613072 \nu^{8} + 96059841589747052611 \nu^{7} - 129712942536712800344 \nu^{6} - 240890934966007223645 \nu^{5} + 280741064934462981181 \nu^{4} + 272973521847255931684 \nu^{3} - 285212960447565670230 \nu^{2} - 88113331148205643228 \nu + 88599863268713580968\)\()/ 7840883481437704 \)
\(\beta_{14}\)\(=\)\((\)\(12507240909772965 \nu^{15} - 33646326659730001 \nu^{14} - 527725179776986492 \nu^{13} + 1231914240821691860 \nu^{12} + 9063391346572318730 \nu^{11} - 17832062238083640769 \nu^{10} - 80401614076340790046 \nu^{9} + 131346321914925220744 \nu^{8} + 386326816150958387783 \nu^{7} - 529515818232642909092 \nu^{6} - 968407126713045024321 \nu^{5} + 1147728394339846755949 \nu^{4} + 1096868025231990569216 \nu^{3} - 1167474063125882771310 \nu^{2} - 353504367066187383884 \nu + 363101975930316935296\)\()/ 15681766962875408 \)
\(\beta_{15}\)\(=\)\((\)\(3695268078821707 \nu^{15} - 9895392816158751 \nu^{14} - 156009228278353240 \nu^{13} + 362058147223910800 \nu^{12} + 2680768089249478598 \nu^{11} - 5235830759322990715 \nu^{10} - 23790467262191524218 \nu^{9} + 38519889889014706292 \nu^{8} + 114339342138814438977 \nu^{7} - 155097830554963060684 \nu^{6} - 286655332333952125587 \nu^{5} + 335808992886363284979 \nu^{4} + 324775010444080001192 \nu^{3} - 341269633737372277150 \nu^{2} - 104837318910714038052 \nu + 106042702582046298700\)\()/ 3920441740718852 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{12} + \beta_{11} + 6\)
\(\nu^{3}\)\(=\)\(-\beta_{12} + \beta_{11} - \beta_{8} - \beta_{5} - \beta_{4} - \beta_{2} + 9 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{13} - 13 \beta_{12} + 14 \beta_{11} + \beta_{10} - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 5 \beta_{1} + 53\)
\(\nu^{5}\)\(=\)\(-2 \beta_{15} + 2 \beta_{14} + \beta_{13} - 20 \beta_{12} + 23 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} - 15 \beta_{8} - \beta_{7} + 3 \beta_{6} - 15 \beta_{5} - 18 \beta_{4} + 2 \beta_{3} - 20 \beta_{2} + 95 \beta_{1} + 32\)
\(\nu^{6}\)\(=\)\(-6 \beta_{15} + 26 \beta_{13} - 166 \beta_{12} + 185 \beta_{11} + 22 \beta_{10} - 23 \beta_{9} - 5 \beta_{8} + 17 \beta_{7} + 20 \beta_{6} - 10 \beta_{5} - 18 \beta_{4} + 47 \beta_{3} - 67 \beta_{2} + 116 \beta_{1} + 557\)
\(\nu^{7}\)\(=\)\(-53 \beta_{15} + 37 \beta_{14} + 44 \beta_{13} - 338 \beta_{12} + 400 \beta_{11} + 72 \beta_{10} - 73 \beta_{9} - 203 \beta_{8} - 27 \beta_{7} + 81 \beta_{6} - 215 \beta_{5} - 248 \beta_{4} + 32 \beta_{3} - 325 \beta_{2} + 1105 \beta_{1} + 636\)
\(\nu^{8}\)\(=\)\(-179 \beta_{15} + \beta_{14} + 493 \beta_{13} - 2180 \beta_{12} + 2471 \beta_{11} + 385 \beta_{10} - 389 \beta_{9} - 159 \beta_{8} + 189 \beta_{7} + 332 \beta_{6} - 295 \beta_{5} - 277 \beta_{4} + 573 \beta_{3} - 1136 \beta_{2} + 2012 \beta_{1} + 6477\)
\(\nu^{9}\)\(=\)\(-1042 \beta_{15} + 504 \beta_{14} + 1108 \beta_{13} - 5323 \beta_{12} + 6331 \beta_{11} + 1308 \beta_{10} - 1351 \beta_{9} - 2730 \beta_{8} - 555 \beta_{7} + 1529 \beta_{6} - 3096 \beta_{5} - 3176 \beta_{4} + 371 \beta_{3} - 4909 \beta_{2} + 13660 \beta_{1} + 10766\)
\(\nu^{10}\)\(=\)\(-3702 \beta_{15} + 15 \beta_{14} + 8295 \beta_{13} - 29297 \beta_{12} + 33572 \beta_{11} + 6249 \beta_{10} - 6111 \beta_{9} - 3424 \beta_{8} + 1449 \beta_{7} + 5309 \beta_{6} - 6093 \beta_{5} - 4157 \beta_{4} + 6437 \beta_{3} - 17620 \beta_{2} + 31504 \beta_{1} + 80154\)
\(\nu^{11}\)\(=\)\(-18139 \beta_{15} + 6057 \beta_{14} + 22118 \beta_{13} - 80811 \beta_{12} + 96211 \beta_{11} + 21659 \beta_{10} - 22791 \beta_{9} - 36925 \beta_{8} - 10278 \beta_{7} + 25248 \beta_{6} - 44809 \beta_{5} - 39894 \beta_{4} + 3705 \beta_{3} - 72142 \beta_{2} + 175835 \beta_{1} + 169350\)
\(\nu^{12}\)\(=\)\(-65998 \beta_{15} - 242 \beta_{14} + 131574 \beta_{13} - 400190 \beta_{12} + 462969 \beta_{11} + 97786 \beta_{10} - 94813 \beta_{9} - 62592 \beta_{8} + 1478 \beta_{7} + 83562 \beta_{6} - 109098 \beta_{5} - 61819 \beta_{4} + 69544 \beta_{3} - 263646 \beta_{2} + 471316 \beta_{1} + 1032336\)
\(\nu^{13}\)\(=\)\(-295606 \beta_{15} + 66812 \beta_{14} + 391243 \beta_{13} - 1201340 \beta_{12} + 1433236 \beta_{11} + 344263 \beta_{10} - 369200 \beta_{9} - 502714 \beta_{8} - 180397 \beta_{7} + 391880 \beta_{6} - 650273 \beta_{5} - 501775 \beta_{4} + 32620 \beta_{3} - 1049040 \beta_{2} + 2327489 \beta_{1} + 2563372\)
\(\nu^{14}\)\(=\)\(-1088984 \beta_{15} - 17948 \beta_{14} + 2018086 \beta_{13} - 5529279 \beta_{12} + 6459587 \beta_{11} + 1498058 \beta_{10} - 1473675 \beta_{9} - 1048202 \beta_{8} - 242986 \beta_{7} + 1296392 \beta_{6} - 1813802 \beta_{5} - 912149 \beta_{4} + 732765 \beta_{3} - 3880474 \beta_{2} + 6894287 \beta_{1} + 13659793\)
\(\nu^{15}\)\(=\)\(-4625695 \beta_{15} + 668168 \beta_{14} + 6448737 \beta_{13} - 17630107 \beta_{12} + 21122914 \beta_{11} + 5351563 \beta_{10} - 5859785 \beta_{9} - 6884898 \beta_{8} - 3065032 \beta_{7} + 5892818 \beta_{6} - 9447603 \beta_{5} - 6377791 \beta_{4} + 238909 \beta_{3} - 15188046 \beta_{2} + 31424408 \beta_{1} + 37965894\)

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
−3.26657
−2.86082
−2.68392
−2.44876
−2.07316
−1.52621
−0.719032
0.771393
0.808021
1.64745
1.78856
2.32389
2.32411
3.35142
3.75055
3.81307
1.00000 1.00000 1.00000 −3.26657 1.00000 1.36909 1.00000 1.00000 −3.26657
1.2 1.00000 1.00000 1.00000 −2.86082 1.00000 2.85228 1.00000 1.00000 −2.86082
1.3 1.00000 1.00000 1.00000 −2.68392 1.00000 −4.10513 1.00000 1.00000 −2.68392
1.4 1.00000 1.00000 1.00000 −2.44876 1.00000 −1.82729 1.00000 1.00000 −2.44876
1.5 1.00000 1.00000 1.00000 −2.07316 1.00000 −0.689093 1.00000 1.00000 −2.07316
1.6 1.00000 1.00000 1.00000 −1.52621 1.00000 4.59536 1.00000 1.00000 −1.52621
1.7 1.00000 1.00000 1.00000 −0.719032 1.00000 −5.05986 1.00000 1.00000 −0.719032
1.8 1.00000 1.00000 1.00000 0.771393 1.00000 −0.402524 1.00000 1.00000 0.771393
1.9 1.00000 1.00000 1.00000 0.808021 1.00000 3.24300 1.00000 1.00000 0.808021
1.10 1.00000 1.00000 1.00000 1.64745 1.00000 3.14335 1.00000 1.00000 1.64745
1.11 1.00000 1.00000 1.00000 1.78856 1.00000 0.327596 1.00000 1.00000 1.78856
1.12 1.00000 1.00000 1.00000 2.32389 1.00000 −0.374750 1.00000 1.00000 2.32389
1.13 1.00000 1.00000 1.00000 2.32411 1.00000 −3.70744 1.00000 1.00000 2.32411
1.14 1.00000 1.00000 1.00000 3.35142 1.00000 2.91939 1.00000 1.00000 3.35142
1.15 1.00000 1.00000 1.00000 3.75055 1.00000 −2.75149 1.00000 1.00000 3.75055
1.16 1.00000 1.00000 1.00000 3.81307 1.00000 4.46750 1.00000 1.00000 3.81307
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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 8034.2.a.bd 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.bd 16 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(1\)
\(103\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{16} - \cdots\)
\(T_{7}^{16} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{16} \)
$3$ \( ( 1 - T )^{16} \)
$5$ \( 1 - 5 T + 44 T^{2} - 179 T^{3} + 978 T^{4} - 3486 T^{5} + 14830 T^{6} - 47362 T^{7} + 170463 T^{8} - 494728 T^{9} + 1569001 T^{10} - 4172967 T^{11} + 11922100 T^{12} - 29207576 T^{13} + 76147049 T^{14} - 172205237 T^{15} + 412689308 T^{16} - 861026185 T^{17} + 1903676225 T^{18} - 3650947000 T^{19} + 7451312500 T^{20} - 13040521875 T^{21} + 24515640625 T^{22} - 38650625000 T^{23} + 66587109375 T^{24} - 92503906250 T^{25} + 144824218750 T^{26} - 170214843750 T^{27} + 238769531250 T^{28} - 218505859375 T^{29} + 268554687500 T^{30} - 152587890625 T^{31} + 152587890625 T^{32} \)
$7$ \( 1 - 4 T + 46 T^{2} - 143 T^{3} + 1004 T^{4} - 2510 T^{5} + 14304 T^{6} - 29403 T^{7} + 154340 T^{8} - 258336 T^{9} + 1366553 T^{10} - 1787711 T^{11} + 10480872 T^{12} - 10197411 T^{13} + 73931067 T^{14} - 55711752 T^{15} + 513067434 T^{16} - 389982264 T^{17} + 3622622283 T^{18} - 3497711973 T^{19} + 25164573672 T^{20} - 30046058777 T^{21} + 160773593897 T^{22} - 212750804448 T^{23} + 889739386340 T^{24} - 1186517106621 T^{25} + 4040525961696 T^{26} - 4963090124930 T^{27} + 13896652349804 T^{28} - 13855128488201 T^{29} + 31198261351054 T^{30} - 18990246039772 T^{31} + 33232930569601 T^{32} \)
$11$ \( 1 - 18 T + 222 T^{2} - 1987 T^{3} + 14824 T^{4} - 93992 T^{5} + 532681 T^{6} - 2730734 T^{7} + 13026106 T^{8} - 58197357 T^{9} + 247564285 T^{10} - 1003286131 T^{11} + 3908767024 T^{12} - 14608731198 T^{13} + 52737399316 T^{14} - 183510103533 T^{15} + 619164682490 T^{16} - 2018611138863 T^{17} + 6381225317236 T^{18} - 19444221224538 T^{19} + 57228257998384 T^{20} - 161580234683681 T^{21} + 438575232298885 T^{22} - 1134101847607047 T^{23} + 2792261505947386 T^{24} - 6438927930035194 T^{25} + 13816373273885281 T^{26} - 26817014544069112 T^{27} + 46524062256512104 T^{28} - 68596629029990897 T^{29} + 84304463055479502 T^{30} - 75190467049481718 T^{31} + 45949729863572161 T^{32} \)
$13$ \( ( 1 + T )^{16} \)
$17$ \( 1 - 17 T + 239 T^{2} - 2365 T^{3} + 20934 T^{4} - 157674 T^{5} + 1101441 T^{6} - 6967309 T^{7} + 41698740 T^{8} - 232884971 T^{9} + 1247156179 T^{10} - 6334270028 T^{11} + 31065016074 T^{12} - 145589136725 T^{13} + 660576422525 T^{14} - 2872258232223 T^{15} + 12094789738742 T^{16} - 48828389947791 T^{17} + 190906586109725 T^{18} - 715279428729925 T^{19} + 2594581207516554 T^{20} - 8993757639145996 T^{21} + 30103318324388851 T^{22} - 95561709961783483 T^{23} + 290880295835324340 T^{24} - 826238379208436573 T^{25} + 2220498337704447009 T^{26} - 5403786978409725642 T^{27} + 12196613914167816774 T^{28} - 23424327047822541005 T^{29} + 40242300547696822031 T^{30} - 48661191875666868481 T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 - 8 T + 174 T^{2} - 1168 T^{3} + 14381 T^{4} - 86472 T^{5} + 778232 T^{6} - 4352887 T^{7} + 31417374 T^{8} - 166970461 T^{9} + 1014611644 T^{10} - 5179103620 T^{11} + 27393112675 T^{12} - 134398590804 T^{13} + 635857427246 T^{14} - 2972024352442 T^{15} + 12892474396930 T^{16} - 56468462696398 T^{17} + 229544531235806 T^{18} - 921839934324636 T^{19} + 3569897836918675 T^{20} - 12823973294378380 T^{21} + 47733298664838364 T^{22} - 149250176335701679 T^{23} + 533578951911674334 T^{24} - 1404623084722137973 T^{25} + 4771391955940987832 T^{26} - 10073145667446793368 T^{27} + 31829681851090461341 T^{28} - 49117884683916244912 T^{29} + \)\(13\!\cdots\!54\)\( T^{30} - \)\(12\!\cdots\!92\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( 1 - 9 T + 215 T^{2} - 1397 T^{3} + 20893 T^{4} - 106801 T^{5} + 1308494 T^{6} - 5434270 T^{7} + 60807819 T^{8} - 206840189 T^{9} + 2259169676 T^{10} - 6331804517 T^{11} + 70594666072 T^{12} - 166376157997 T^{13} + 1922330798959 T^{14} - 3995384062124 T^{15} + 46645041506046 T^{16} - 91893833428852 T^{17} + 1016912992649311 T^{18} - 2024298714349499 T^{19} + 19755281948254552 T^{20} - 40753665680361331 T^{21} + 334438191388501964 T^{22} - 704254738969489483 T^{23} + 4761920218678712139 T^{24} - 9787949873608537010 T^{25} + 54206341363992434606 T^{26} - \)\(10\!\cdots\!27\)\( T^{27} + \)\(45\!\cdots\!53\)\( T^{28} - \)\(70\!\cdots\!51\)\( T^{29} + \)\(24\!\cdots\!35\)\( T^{30} - \)\(23\!\cdots\!63\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( 1 - 14 T + 363 T^{2} - 3784 T^{3} + 57687 T^{4} - 485905 T^{5} + 5596325 T^{6} - 39845926 T^{7} + 382677987 T^{8} - 2374133520 T^{9} + 20075682866 T^{10} - 111107231470 T^{11} + 855577268701 T^{12} - 4306759012511 T^{13} + 30791526054198 T^{14} - 143152941955934 T^{15} + 957227518695760 T^{16} - 4151435316722086 T^{17} + 25895673411580518 T^{18} - 105037545556130779 T^{19} + 605133546184111981 T^{20} - 2278936979658659030 T^{21} + 11941484353696917986 T^{22} - 40953509561050777680 T^{23} + \)\(19\!\cdots\!07\)\( T^{24} - \)\(57\!\cdots\!94\)\( T^{25} + \)\(23\!\cdots\!25\)\( T^{26} - \)\(59\!\cdots\!45\)\( T^{27} + \)\(20\!\cdots\!67\)\( T^{28} - \)\(38\!\cdots\!76\)\( T^{29} + \)\(10\!\cdots\!03\)\( T^{30} - \)\(12\!\cdots\!86\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( 1 - 12 T + 242 T^{2} - 1868 T^{3} + 23811 T^{4} - 134798 T^{5} + 1453454 T^{6} - 6071611 T^{7} + 64167772 T^{8} - 170180139 T^{9} + 2164384854 T^{10} - 1609886068 T^{11} + 58820785357 T^{12} + 108118581150 T^{13} + 1425717210538 T^{14} + 6884627377524 T^{15} + 38315233057190 T^{16} + 213423448703244 T^{17} + 1370114239327018 T^{18} + 3220960651039650 T^{19} + 54322230513681997 T^{20} - 46089671333568268 T^{21} + 1920899525025647574 T^{22} - 4682100493663341429 T^{23} + 54728117631357551452 T^{24} - \)\(16\!\cdots\!81\)\( T^{25} + \)\(11\!\cdots\!54\)\( T^{26} - \)\(34\!\cdots\!38\)\( T^{27} + \)\(18\!\cdots\!71\)\( T^{28} - \)\(45\!\cdots\!88\)\( T^{29} + \)\(18\!\cdots\!82\)\( T^{30} - \)\(28\!\cdots\!12\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( 1 - 31 T + 754 T^{2} - 12946 T^{3} + 190736 T^{4} - 2348853 T^{5} + 25831880 T^{6} - 251058916 T^{7} + 2228836341 T^{8} - 17973740021 T^{9} + 134598123464 T^{10} - 933972070364 T^{11} + 6132901071232 T^{12} - 38247599798141 T^{13} + 232742218915806 T^{14} - 1394690371823392 T^{15} + 8452192989535540 T^{16} - 51603543757465504 T^{17} + 318624097695738414 T^{18} - 1937355672575236073 T^{19} + 11494044004561236352 T^{20} - 64765319086522190348 T^{21} + \)\(34\!\cdots\!76\)\( T^{22} - \)\(17\!\cdots\!93\)\( T^{23} + \)\(78\!\cdots\!61\)\( T^{24} - \)\(32\!\cdots\!32\)\( T^{25} + \)\(12\!\cdots\!20\)\( T^{26} - \)\(41\!\cdots\!89\)\( T^{27} + \)\(12\!\cdots\!16\)\( T^{28} - \)\(31\!\cdots\!62\)\( T^{29} + \)\(67\!\cdots\!06\)\( T^{30} - \)\(10\!\cdots\!83\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( 1 - 29 T + 705 T^{2} - 11962 T^{3} + 178582 T^{4} - 2236960 T^{5} + 25459115 T^{6} - 258955312 T^{7} + 2448304670 T^{8} - 21381960159 T^{9} + 176773823974 T^{10} - 1379175637927 T^{11} + 10334227830435 T^{12} - 74086484916755 T^{13} + 514886627051218 T^{14} - 3447544393960184 T^{15} + 22468553587821000 T^{16} - 141349320152367544 T^{17} + 865524420073097458 T^{18} - 5106114626947671355 T^{19} + 29202057968357836035 T^{20} - \)\(15\!\cdots\!27\)\( T^{21} + \)\(83\!\cdots\!34\)\( T^{22} - \)\(41\!\cdots\!79\)\( T^{23} + \)\(19\!\cdots\!70\)\( T^{24} - \)\(84\!\cdots\!32\)\( T^{25} + \)\(34\!\cdots\!15\)\( T^{26} - \)\(12\!\cdots\!60\)\( T^{27} + \)\(40\!\cdots\!42\)\( T^{28} - \)\(11\!\cdots\!02\)\( T^{29} + \)\(26\!\cdots\!05\)\( T^{30} - \)\(45\!\cdots\!29\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( 1 - 30 T + 753 T^{2} - 13538 T^{3} + 211812 T^{4} - 2841218 T^{5} + 34443732 T^{6} - 379159713 T^{7} + 3871789827 T^{8} - 36847112340 T^{9} + 330610251346 T^{10} - 2801607814827 T^{11} + 22585491182096 T^{12} - 173209703194475 T^{13} + 1269592396734257 T^{14} - 8888276249156005 T^{15} + 59596840617230128 T^{16} - 382195878713708215 T^{17} + 2347476341561641193 T^{18} - 13771383871883123825 T^{19} + 77215299838840986896 T^{20} - \)\(41\!\cdots\!61\)\( T^{21} + \)\(20\!\cdots\!54\)\( T^{22} - \)\(10\!\cdots\!80\)\( T^{23} + \)\(45\!\cdots\!27\)\( T^{24} - \)\(19\!\cdots\!59\)\( T^{25} + \)\(74\!\cdots\!68\)\( T^{26} - \)\(26\!\cdots\!26\)\( T^{27} + \)\(84\!\cdots\!12\)\( T^{28} - \)\(23\!\cdots\!34\)\( T^{29} + \)\(55\!\cdots\!97\)\( T^{30} - \)\(95\!\cdots\!10\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 + T + 198 T^{2} + 582 T^{3} + 20803 T^{4} + 53517 T^{5} + 1765785 T^{6} + 1234928 T^{7} + 113994985 T^{8} - 67893476 T^{9} + 5834256239 T^{10} - 14609562287 T^{11} + 297660346061 T^{12} - 1395538072257 T^{13} + 13891813886354 T^{14} - 77951950606532 T^{15} + 604027334011596 T^{16} - 3663741678507004 T^{17} + 30687016874955986 T^{18} - 144888949275938511 T^{19} + 1452487535127286541 T^{20} - 3350630164978951009 T^{21} + 62888704284742687631 T^{22} - 34396404670199799388 T^{23} + \)\(27\!\cdots\!85\)\( T^{24} + \)\(13\!\cdots\!76\)\( T^{25} + \)\(92\!\cdots\!65\)\( T^{26} + \)\(13\!\cdots\!51\)\( T^{27} + \)\(24\!\cdots\!23\)\( T^{28} + \)\(31\!\cdots\!14\)\( T^{29} + \)\(50\!\cdots\!62\)\( T^{30} + \)\(12\!\cdots\!43\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 - 12 T + 568 T^{2} - 5149 T^{3} + 145525 T^{4} - 1046110 T^{5} + 23545152 T^{6} - 139497358 T^{7} + 2803144996 T^{8} - 14169466826 T^{9} + 266231223656 T^{10} - 1181358941884 T^{11} + 20997704216835 T^{12} - 83575642193485 T^{13} + 1402678955948192 T^{14} - 5096652195214124 T^{15} + 80205921351636774 T^{16} - 270122566346348572 T^{17} + 3940125187258471328 T^{18} - 12442490882839466345 T^{19} + \)\(16\!\cdots\!35\)\( T^{20} - \)\(49\!\cdots\!12\)\( T^{21} + \)\(59\!\cdots\!24\)\( T^{22} - \)\(16\!\cdots\!62\)\( T^{23} + \)\(17\!\cdots\!56\)\( T^{24} - \)\(46\!\cdots\!14\)\( T^{25} + \)\(41\!\cdots\!48\)\( T^{26} - \)\(96\!\cdots\!70\)\( T^{27} + \)\(71\!\cdots\!25\)\( T^{28} - \)\(13\!\cdots\!77\)\( T^{29} + \)\(78\!\cdots\!92\)\( T^{30} - \)\(87\!\cdots\!84\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 - 38 T + 1095 T^{2} - 23297 T^{3} + 424605 T^{4} - 6672795 T^{5} + 94580432 T^{6} - 1217058846 T^{7} + 14511574567 T^{8} - 161041405813 T^{9} + 1682485451586 T^{10} - 16594154775321 T^{11} + 155630262017655 T^{12} - 1389923566921313 T^{13} + 11876680328710591 T^{14} - 97108684461423523 T^{15} + 761932837815421832 T^{16} - 5729412383223987857 T^{17} + 41342724224241567271 T^{18} - \)\(28\!\cdots\!27\)\( T^{19} + \)\(18\!\cdots\!55\)\( T^{20} - \)\(11\!\cdots\!79\)\( T^{21} + \)\(70\!\cdots\!26\)\( T^{22} - \)\(40\!\cdots\!47\)\( T^{23} + \)\(21\!\cdots\!07\)\( T^{24} - \)\(10\!\cdots\!94\)\( T^{25} + \)\(48\!\cdots\!32\)\( T^{26} - \)\(20\!\cdots\!05\)\( T^{27} + \)\(75\!\cdots\!05\)\( T^{28} - \)\(24\!\cdots\!63\)\( T^{29} + \)\(67\!\cdots\!95\)\( T^{30} - \)\(13\!\cdots\!62\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 + 459 T^{2} - 318 T^{3} + 103898 T^{4} - 159398 T^{5} + 15704366 T^{6} - 38331037 T^{7} + 1814368519 T^{8} - 5937657768 T^{9} + 173417385168 T^{10} - 671036989903 T^{11} + 14346471436314 T^{12} - 59441130966933 T^{13} + 1047651981169903 T^{14} - 4318613747814279 T^{15} + 67852682743911080 T^{16} - 263435438616671019 T^{17} + 3898313021933209063 T^{18} - 13492007348005419273 T^{19} + \)\(19\!\cdots\!74\)\( T^{20} - \)\(56\!\cdots\!03\)\( T^{21} + \)\(89\!\cdots\!48\)\( T^{22} - \)\(18\!\cdots\!28\)\( T^{23} + \)\(34\!\cdots\!39\)\( T^{24} - \)\(44\!\cdots\!17\)\( T^{25} + \)\(11\!\cdots\!66\)\( T^{26} - \)\(69\!\cdots\!78\)\( T^{27} + \)\(27\!\cdots\!58\)\( T^{28} - \)\(51\!\cdots\!58\)\( T^{29} + \)\(45\!\cdots\!19\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( 1 - 28 T + 858 T^{2} - 15649 T^{3} + 292080 T^{4} - 4225007 T^{5} + 61602204 T^{6} - 771198098 T^{7} + 9574930223 T^{8} - 107511352491 T^{9} + 1179390780967 T^{10} - 12081186344489 T^{11} + 119596924133268 T^{12} - 1127862817133331 T^{13} + 10201063163519899 T^{14} - 88868275954132921 T^{15} + 739804228787108344 T^{16} - 5954174488926905707 T^{17} + 45792572541040826611 T^{18} - \)\(33\!\cdots\!53\)\( T^{19} + \)\(24\!\cdots\!28\)\( T^{20} - \)\(16\!\cdots\!23\)\( T^{21} + \)\(10\!\cdots\!23\)\( T^{22} - \)\(65\!\cdots\!93\)\( T^{23} + \)\(38\!\cdots\!43\)\( T^{24} - \)\(20\!\cdots\!06\)\( T^{25} + \)\(11\!\cdots\!96\)\( T^{26} - \)\(51\!\cdots\!81\)\( T^{27} + \)\(23\!\cdots\!80\)\( T^{28} - \)\(85\!\cdots\!63\)\( T^{29} + \)\(31\!\cdots\!82\)\( T^{30} - \)\(68\!\cdots\!04\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( 1 - 32 T + 995 T^{2} - 20508 T^{3} + 395622 T^{4} - 6259188 T^{5} + 93755982 T^{6} - 1239096885 T^{7} + 15716903928 T^{8} - 182294906860 T^{9} + 2049415143507 T^{10} - 21476154775115 T^{11} + 219328591407162 T^{12} - 2107956202438259 T^{13} + 19784162377688964 T^{14} - 175455499172719087 T^{15} + 1520075125513646654 T^{16} - 12457340441263055177 T^{17} + 99731962545930067524 T^{18} - \)\(75\!\cdots\!49\)\( T^{19} + \)\(55\!\cdots\!22\)\( T^{20} - \)\(38\!\cdots\!65\)\( T^{21} + \)\(26\!\cdots\!47\)\( T^{22} - \)\(16\!\cdots\!60\)\( T^{23} + \)\(10\!\cdots\!08\)\( T^{24} - \)\(56\!\cdots\!35\)\( T^{25} + \)\(30\!\cdots\!82\)\( T^{26} - \)\(14\!\cdots\!48\)\( T^{27} + \)\(64\!\cdots\!02\)\( T^{28} - \)\(23\!\cdots\!88\)\( T^{29} + \)\(82\!\cdots\!95\)\( T^{30} - \)\(18\!\cdots\!32\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \)
$73$ \( 1 - 20 T + 513 T^{2} - 9247 T^{3} + 159213 T^{4} - 2321194 T^{5} + 33317650 T^{6} - 424662675 T^{7} + 5258202404 T^{8} - 60372874351 T^{9} + 673283949509 T^{10} - 7023205288675 T^{11} + 71674972251759 T^{12} - 691734325623476 T^{13} + 6508014098139040 T^{14} - 58294144897789706 T^{15} + 511105999549900974 T^{16} - 4255472577538648538 T^{17} + 34681207128982944160 T^{18} - \)\(26\!\cdots\!92\)\( T^{19} + \)\(20\!\cdots\!19\)\( T^{20} - \)\(14\!\cdots\!75\)\( T^{21} + \)\(10\!\cdots\!01\)\( T^{22} - \)\(66\!\cdots\!47\)\( T^{23} + \)\(42\!\cdots\!24\)\( T^{24} - \)\(25\!\cdots\!75\)\( T^{25} + \)\(14\!\cdots\!50\)\( T^{26} - \)\(72\!\cdots\!38\)\( T^{27} + \)\(36\!\cdots\!73\)\( T^{28} - \)\(15\!\cdots\!51\)\( T^{29} + \)\(62\!\cdots\!17\)\( T^{30} - \)\(17\!\cdots\!40\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 - 13 T + 775 T^{2} - 8945 T^{3} + 289929 T^{4} - 3056030 T^{5} + 70650878 T^{6} - 693207671 T^{7} + 12711402618 T^{8} - 117556710631 T^{9} + 1809285303132 T^{10} - 15865666579748 T^{11} + 212444210262071 T^{12} - 1764614755495823 T^{13} + 21101642992512959 T^{14} - 164760112592133413 T^{15} + 1796776883123990538 T^{16} - 13016048894778539627 T^{17} + \)\(13\!\cdots\!19\)\( T^{18} - \)\(87\!\cdots\!97\)\( T^{19} + \)\(82\!\cdots\!51\)\( T^{20} - \)\(48\!\cdots\!52\)\( T^{21} + \)\(43\!\cdots\!72\)\( T^{22} - \)\(22\!\cdots\!29\)\( T^{23} + \)\(19\!\cdots\!98\)\( T^{24} - \)\(83\!\cdots\!49\)\( T^{25} + \)\(66\!\cdots\!78\)\( T^{26} - \)\(22\!\cdots\!70\)\( T^{27} + \)\(17\!\cdots\!89\)\( T^{28} - \)\(41\!\cdots\!55\)\( T^{29} + \)\(28\!\cdots\!75\)\( T^{30} - \)\(37\!\cdots\!87\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 - 39 T + 1238 T^{2} - 27953 T^{3} + 559862 T^{4} - 9568089 T^{5} + 150555772 T^{6} - 2148033125 T^{7} + 28809842394 T^{8} - 360796910182 T^{9} + 4303964232376 T^{10} - 48724320977567 T^{11} + 529549317747542 T^{12} - 5508676566066797 T^{13} + 55217376675221526 T^{14} - 531723737916086202 T^{15} + 4939932734933838930 T^{16} - 44133070247035154766 T^{17} + \)\(38\!\cdots\!14\)\( T^{18} - \)\(31\!\cdots\!39\)\( T^{19} + \)\(25\!\cdots\!82\)\( T^{20} - \)\(19\!\cdots\!81\)\( T^{21} + \)\(14\!\cdots\!44\)\( T^{22} - \)\(97\!\cdots\!14\)\( T^{23} + \)\(64\!\cdots\!54\)\( T^{24} - \)\(40\!\cdots\!75\)\( T^{25} + \)\(23\!\cdots\!28\)\( T^{26} - \)\(12\!\cdots\!63\)\( T^{27} + \)\(59\!\cdots\!82\)\( T^{28} - \)\(24\!\cdots\!39\)\( T^{29} + \)\(91\!\cdots\!02\)\( T^{30} - \)\(23\!\cdots\!73\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 - 9 T + 685 T^{2} - 5312 T^{3} + 236340 T^{4} - 1586649 T^{5} + 55425874 T^{6} - 328282965 T^{7} + 10036323760 T^{8} - 53526070565 T^{9} + 1494977776540 T^{10} - 7283797771011 T^{11} + 189005283129452 T^{12} - 849011488066604 T^{13} + 20626626362065045 T^{14} - 86079699007121821 T^{15} + 1962582005966434910 T^{16} - 7661093211633842069 T^{17} + \)\(16\!\cdots\!45\)\( T^{18} - \)\(59\!\cdots\!76\)\( T^{19} + \)\(11\!\cdots\!32\)\( T^{20} - \)\(40\!\cdots\!39\)\( T^{21} + \)\(74\!\cdots\!40\)\( T^{22} - \)\(23\!\cdots\!85\)\( T^{23} + \)\(39\!\cdots\!60\)\( T^{24} - \)\(11\!\cdots\!85\)\( T^{25} + \)\(17\!\cdots\!74\)\( T^{26} - \)\(44\!\cdots\!61\)\( T^{27} + \)\(58\!\cdots\!40\)\( T^{28} - \)\(11\!\cdots\!28\)\( T^{29} + \)\(13\!\cdots\!85\)\( T^{30} - \)\(15\!\cdots\!41\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( 1 - 35 T + 1528 T^{2} - 38061 T^{3} + 988783 T^{4} - 19478483 T^{5} + 381496493 T^{6} - 6282347494 T^{7} + 101281076724 T^{8} - 1443044226028 T^{9} + 20048497517841 T^{10} - 253126336351305 T^{11} + 3118810776277033 T^{12} - 35509515023167295 T^{13} + 395496750669768458 T^{14} - 4108810659268075107 T^{15} + 41835419836014023142 T^{16} - \)\(39\!\cdots\!79\)\( T^{17} + \)\(37\!\cdots\!22\)\( T^{18} - \)\(32\!\cdots\!35\)\( T^{19} + \)\(27\!\cdots\!73\)\( T^{20} - \)\(21\!\cdots\!85\)\( T^{21} + \)\(16\!\cdots\!89\)\( T^{22} - \)\(11\!\cdots\!64\)\( T^{23} + \)\(79\!\cdots\!64\)\( T^{24} - \)\(47\!\cdots\!98\)\( T^{25} + \)\(28\!\cdots\!57\)\( T^{26} - \)\(13\!\cdots\!99\)\( T^{27} + \)\(68\!\cdots\!03\)\( T^{28} - \)\(25\!\cdots\!97\)\( T^{29} + \)\(99\!\cdots\!32\)\( T^{30} - \)\(22\!\cdots\!55\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \)
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