# 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

# Learn more about

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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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