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:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 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

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

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{16} \)
$3$ \( ( -1 + T )^{16} \)
$5$ \( -66432 + 94528 T + 185524 T^{2} - 296326 T^{3} - 122680 T^{4} + 270913 T^{5} + 19731 T^{6} - 113788 T^{7} + 6763 T^{8} + 25368 T^{9} - 3150 T^{10} - 3101 T^{11} + 498 T^{12} + 196 T^{13} - 36 T^{14} - 5 T^{15} + T^{16} \)
$7$ \( 31456 + 84352 T - 301910 T^{2} - 921151 T^{3} + 136818 T^{4} + 1185952 T^{5} - 137971 T^{6} - 487663 T^{7} + 80924 T^{8} + 85712 T^{9} - 17210 T^{10} - 7137 T^{11} + 1592 T^{12} + 277 T^{13} - 66 T^{14} - 4 T^{15} + T^{16} \)
$11$ \( -134528 + 1508960 T - 3492976 T^{2} - 7211144 T^{3} + 19382888 T^{4} - 9331906 T^{5} - 5309820 T^{6} + 5712533 T^{7} - 1096948 T^{8} - 397007 T^{9} + 175775 T^{10} - 5871 T^{11} - 6780 T^{12} + 983 T^{13} + 46 T^{14} - 18 T^{15} + T^{16} \)
$13$ \( ( 1 + T )^{16} \)
$17$ \( 2278916096 - 3188285440 T + 365424896 T^{2} + 1310048064 T^{3} - 511732288 T^{4} - 164581680 T^{5} + 112686832 T^{6} + 1797292 T^{7} - 10309836 T^{8} + 1059479 T^{9} + 419996 T^{10} - 77179 T^{11} - 5892 T^{12} + 1970 T^{13} - 33 T^{14} - 17 T^{15} + T^{16} \)
$19$ \( 38610944 - 121735680 T + 74435072 T^{2} + 127861824 T^{3} - 185941536 T^{4} + 54566080 T^{5} + 27295416 T^{6} - 15538960 T^{7} - 719492 T^{8} + 1402441 T^{9} - 78326 T^{10} - 57896 T^{11} + 5641 T^{12} + 1112 T^{13} - 130 T^{14} - 8 T^{15} + T^{16} \)
$23$ \( -69392896 - 605799096 T + 772090996 T^{2} + 1392145590 T^{3} - 1428393047 T^{4} + 155010833 T^{5} + 189849832 T^{6} - 49623781 T^{7} - 7961353 T^{8} + 3663863 T^{9} + 16837 T^{10} - 117588 T^{11} + 6679 T^{12} + 1708 T^{13} - 153 T^{14} - 9 T^{15} + T^{16} \)
$29$ \( 169924096 + 1076872832 T + 1601098456 T^{2} - 78042100 T^{3} - 756423154 T^{4} + 16247053 T^{5} + 145203795 T^{6} - 13721498 T^{7} - 12826855 T^{8} + 2203059 T^{9} + 443112 T^{10} - 118997 T^{11} - 2227 T^{12} + 2306 T^{13} - 101 T^{14} - 14 T^{15} + T^{16} \)
$31$ \( -31423913984 - 31583528448 T + 45550812032 T^{2} + 23151060032 T^{3} - 26233633008 T^{4} + 2880042704 T^{5} + 2157993212 T^{6} - 481676974 T^{7} - 55209694 T^{8} + 21520187 T^{9} + 16604 T^{10} - 419874 T^{11} + 18727 T^{12} + 3712 T^{13} - 254 T^{14} - 12 T^{15} + T^{16} \)
$37$ \( 1639511936 + 970918880 T - 4379594624 T^{2} - 35350920 T^{3} + 3049641020 T^{4} - 750572906 T^{5} - 616796636 T^{6} + 296406877 T^{7} - 12059483 T^{8} - 15258730 T^{9} + 2796642 T^{10} + 58663 T^{11} - 57460 T^{12} + 4259 T^{13} + 162 T^{14} - 31 T^{15} + T^{16} \)
$41$ \( 54262501552 - 56885362720 T - 20773931112 T^{2} + 32871796616 T^{3} - 332592013 T^{4} - 6300151280 T^{5} + 955436173 T^{6} + 453407791 T^{7} - 118081802 T^{8} - 7469307 T^{9} + 4589664 T^{10} - 248624 T^{11} - 51264 T^{12} + 5873 T^{13} + 49 T^{14} - 29 T^{15} + T^{16} \)
$43$ \( 94256128 + 71885056 T - 1101734464 T^{2} - 73058576 T^{3} + 3263158328 T^{4} - 1932690180 T^{5} - 328112084 T^{6} + 496810979 T^{7} - 106467231 T^{8} - 6395379 T^{9} + 4369145 T^{10} - 265776 T^{11} - 49198 T^{12} + 5812 T^{13} + 65 T^{14} - 30 T^{15} + T^{16} \)
$47$ \( -149504384448 - 23102848802 T + 367140646560 T^{2} - 182838736347 T^{3} - 97600556384 T^{4} + 83730366812 T^{5} - 12470990506 T^{6} - 2619071054 T^{7} + 669211813 T^{8} + 28581784 T^{9} - 12834389 T^{10} - 103275 T^{11} + 120255 T^{12} - 123 T^{13} - 554 T^{14} + T^{15} + T^{16} \)
$53$ \( -859031552 + 26811010560 T - 46873717504 T^{2} + 20495983296 T^{3} + 8174739072 T^{4} - 9649567552 T^{5} + 2642748488 T^{6} + 51720968 T^{7} - 152011514 T^{8} + 21548707 T^{9} + 1440972 T^{10} - 532169 T^{11} + 16205 T^{12} + 4391 T^{13} - 280 T^{14} - 12 T^{15} + T^{16} \)
$59$ \( 6488733696 - 20434062208 T - 59324342464 T^{2} + 51836582888 T^{3} + 54779653872 T^{4} - 56055028727 T^{5} + 9787653444 T^{6} + 2393466761 T^{7} - 705622833 T^{8} - 11500046 T^{9} + 15167199 T^{10} - 709016 T^{11} - 117841 T^{12} + 10333 T^{13} + 151 T^{14} - 38 T^{15} + T^{16} \)
$61$ \( 70500352 + 859300352 T - 1259411456 T^{2} - 22716779952 T^{3} + 47257621528 T^{4} - 2118868236 T^{5} - 7145857836 T^{6} + 268876841 T^{7} + 370902461 T^{8} - 8288049 T^{9} - 8734979 T^{10} + 92776 T^{11} + 98896 T^{12} - 318 T^{13} - 517 T^{14} + T^{16} \)
$67$ \( -27355953223296 + 43510750833760 T - 26302438827176 T^{2} + 6548207174566 T^{3} + 84698214082 T^{4} - 387046319637 T^{5} + 62886977007 T^{6} + 3346362563 T^{7} - 1788339457 T^{8} + 92363696 T^{9} + 17471582 T^{10} - 1907008 T^{11} - 45868 T^{12} + 12491 T^{13} - 214 T^{14} - 28 T^{15} + T^{16} \)
$71$ \( 587330945024 - 927908530176 T + 40873660800 T^{2} + 497320872384 T^{3} - 168875659414 T^{4} - 51486588594 T^{5} + 26371163727 T^{6} + 292912031 T^{7} - 1169790174 T^{8} + 79191923 T^{9} + 16917581 T^{10} - 1848384 T^{11} - 69144 T^{12} + 13572 T^{13} - 141 T^{14} - 32 T^{15} + T^{16} \)
$73$ \( -15316577831168 - 17972513400560 T + 10774895515184 T^{2} + 18812703293634 T^{3} + 6288059737742 T^{4} + 134844121773 T^{5} - 230184496623 T^{6} - 20048479176 T^{7} + 3586443712 T^{8} + 375827512 T^{9} - 32586093 T^{10} - 3137991 T^{11} + 189143 T^{12} + 12653 T^{13} - 655 T^{14} - 20 T^{15} + T^{16} \)
$79$ \( 704830767104 - 2013384081408 T + 1570594721792 T^{2} - 228511761152 T^{3} - 189548912736 T^{4} + 62294247584 T^{5} + 5311366336 T^{6} - 3745611140 T^{7} + 72297214 T^{8} + 96423399 T^{9} - 5319867 T^{10} - 1171485 T^{11} + 81843 T^{12} + 6460 T^{13} - 489 T^{14} - 13 T^{15} + T^{16} \)
$83$ \( -81294763022592 - 109978540052800 T + 20359264051584 T^{2} + 21055024959440 T^{3} - 4783209920556 T^{4} - 754244001949 T^{5} + 248305447328 T^{6} + 4022709035 T^{7} - 5210117594 T^{8} + 203113602 T^{9} + 48365010 T^{10} - 3587192 T^{11} - 162238 T^{12} + 20602 T^{13} - 90 T^{14} - 39 T^{15} + T^{16} \)
$89$ \( 27735322681344 + 7276440666112 T - 10095481451648 T^{2} - 2714170771616 T^{3} + 1227934244848 T^{4} + 320280471720 T^{5} - 62306424448 T^{6} - 13474676558 T^{7} + 1798037870 T^{8} + 234881801 T^{9} - 27341189 T^{10} - 1856675 T^{11} + 206614 T^{12} + 6703 T^{13} - 739 T^{14} - 9 T^{15} + T^{16} \)
$97$ \( 487658888192 - 765146446336 T - 211137105792 T^{2} + 425308675904 T^{3} - 39918428416 T^{4} - 60174510544 T^{5} + 13611622392 T^{6} + 2105452684 T^{7} - 812908928 T^{8} + 8134807 T^{9} + 16317489 T^{10} - 1121912 T^{11} - 107705 T^{12} + 12864 T^{13} - 24 T^{14} - 35 T^{15} + T^{16} \)
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