Properties

Label 6024.2.a.q
Level 6024
Weight 2
Character orbit 6024.a
Self dual Yes
Analytic conductor 48.102
Analytic rank 0
Dimension 18
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6024.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1018821776\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{3}\) \( + \beta_{1} q^{5} \) \( -\beta_{14} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \( + \beta_{1} q^{5} \) \( -\beta_{14} q^{7} \) \(+ q^{9}\) \( -\beta_{15} q^{11} \) \( -\beta_{3} q^{13} \) \( + \beta_{1} q^{15} \) \( -\beta_{4} q^{17} \) \( + ( 1 + \beta_{10} ) q^{19} \) \( -\beta_{14} q^{21} \) \( -\beta_{5} q^{23} \) \( + ( 1 + \beta_{2} ) q^{25} \) \(+ q^{27}\) \( + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} + \beta_{16} ) q^{29} \) \( + ( 1 - \beta_{3} - \beta_{13} ) q^{31} \) \( -\beta_{15} q^{33} \) \( + ( 2 + 2 \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{35} \) \( + ( -1 + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{14} - 2 \beta_{15} - \beta_{16} ) q^{37} \) \( -\beta_{3} q^{39} \) \( + ( 1 + \beta_{1} + \beta_{3} + \beta_{5} - \beta_{10} - \beta_{12} + \beta_{13} ) q^{41} \) \( + ( 1 + \beta_{2} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{11} + 2 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} ) q^{43} \) \( + \beta_{1} q^{45} \) \( + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{12} - \beta_{13} + 2 \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{47} \) \( + ( \beta_{1} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} ) q^{49} \) \( -\beta_{4} q^{51} \) \( + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{53} \) \( + ( 2 + \beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{15} - \beta_{17} ) q^{55} \) \( + ( 1 + \beta_{10} ) q^{57} \) \( + ( \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} + \beta_{17} ) q^{59} \) \( + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{14} + 2 \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{61} \) \( -\beta_{14} q^{63} \) \( + ( 2 - \beta_{1} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{12} + \beta_{14} - \beta_{15} + \beta_{17} ) q^{65} \) \( + ( 4 - \beta_{7} - \beta_{8} + \beta_{15} ) q^{67} \) \( -\beta_{5} q^{69} \) \( + ( 3 \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{12} - \beta_{16} - \beta_{17} ) q^{71} \) \( + ( 2 - \beta_{1} - \beta_{2} + \beta_{6} - \beta_{9} + \beta_{10} + \beta_{12} ) q^{73} \) \( + ( 1 + \beta_{2} ) q^{75} \) \( + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{14} - 2 \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{77} \) \( + ( 1 + \beta_{1} + \beta_{4} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{79} \) \(+ q^{81}\) \( + ( 4 - \beta_{1} - 2 \beta_{5} - 2 \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{16} ) q^{83} \) \( + ( 2 + \beta_{3} - \beta_{4} - \beta_{6} - 3 \beta_{7} + \beta_{9} - \beta_{11} - \beta_{13} - \beta_{14} + 3 \beta_{15} + \beta_{16} - \beta_{17} ) q^{85} \) \( + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} + \beta_{16} ) q^{87} \) \( + ( -1 + \beta_{2} + \beta_{4} + 2 \beta_{6} + 5 \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{11} + 2 \beta_{13} + \beta_{14} - 4 \beta_{15} - \beta_{16} + \beta_{17} ) q^{89} \) \( + ( 2 - 2 \beta_{3} - \beta_{4} - \beta_{7} + \beta_{9} - \beta_{11} - 2 \beta_{13} - \beta_{15} - 2 \beta_{16} ) q^{91} \) \( + ( 1 - \beta_{3} - \beta_{13} ) q^{93} \) \( + ( 3 \beta_{1} + 3 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} - 3 \beta_{15} - 2 \beta_{16} ) q^{95} \) \( + ( 2 - \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{17} ) q^{97} \) \( -\beta_{15} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(18q \) \(\mathstrut +\mathstrut 18q^{3} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(18q \) \(\mathstrut +\mathstrut 18q^{3} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut 8q^{11} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut q^{15} \) \(\mathstrut +\mathstrut 2q^{17} \) \(\mathstrut +\mathstrut 21q^{19} \) \(\mathstrut +\mathstrut 7q^{21} \) \(\mathstrut -\mathstrut 2q^{23} \) \(\mathstrut +\mathstrut 25q^{25} \) \(\mathstrut +\mathstrut 18q^{27} \) \(\mathstrut +\mathstrut 5q^{29} \) \(\mathstrut +\mathstrut 23q^{31} \) \(\mathstrut +\mathstrut 8q^{33} \) \(\mathstrut +\mathstrut 17q^{35} \) \(\mathstrut +\mathstrut 15q^{37} \) \(\mathstrut +\mathstrut 2q^{39} \) \(\mathstrut +\mathstrut 20q^{41} \) \(\mathstrut +\mathstrut 37q^{43} \) \(\mathstrut +\mathstrut q^{45} \) \(\mathstrut -\mathstrut q^{47} \) \(\mathstrut +\mathstrut 33q^{49} \) \(\mathstrut +\mathstrut 2q^{51} \) \(\mathstrut +\mathstrut 2q^{53} \) \(\mathstrut +\mathstrut 26q^{55} \) \(\mathstrut +\mathstrut 21q^{57} \) \(\mathstrut +\mathstrut 24q^{59} \) \(\mathstrut +\mathstrut 20q^{61} \) \(\mathstrut +\mathstrut 7q^{63} \) \(\mathstrut +\mathstrut 26q^{65} \) \(\mathstrut +\mathstrut 49q^{67} \) \(\mathstrut -\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut 15q^{71} \) \(\mathstrut +\mathstrut 15q^{73} \) \(\mathstrut +\mathstrut 25q^{75} \) \(\mathstrut +\mathstrut 6q^{77} \) \(\mathstrut +\mathstrut 23q^{79} \) \(\mathstrut +\mathstrut 18q^{81} \) \(\mathstrut +\mathstrut 38q^{83} \) \(\mathstrut +\mathstrut 21q^{85} \) \(\mathstrut +\mathstrut 5q^{87} \) \(\mathstrut +\mathstrut 31q^{89} \) \(\mathstrut +\mathstrut 48q^{91} \) \(\mathstrut +\mathstrut 23q^{93} \) \(\mathstrut +\mathstrut 13q^{95} \) \(\mathstrut +\mathstrut 25q^{97} \) \(\mathstrut +\mathstrut 8q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18}\mathstrut -\mathstrut \) \(x^{17}\mathstrut -\mathstrut \) \(57\) \(x^{16}\mathstrut +\mathstrut \) \(51\) \(x^{15}\mathstrut +\mathstrut \) \(1328\) \(x^{14}\mathstrut -\mathstrut \) \(1116\) \(x^{13}\mathstrut -\mathstrut \) \(16275\) \(x^{12}\mathstrut +\mathstrut \) \(13699\) \(x^{11}\mathstrut +\mathstrut \) \(112394\) \(x^{10}\mathstrut -\mathstrut \) \(101250\) \(x^{9}\mathstrut -\mathstrut \) \(432956\) \(x^{8}\mathstrut +\mathstrut \) \(439806\) \(x^{7}\mathstrut +\mathstrut \) \(844117\) \(x^{6}\mathstrut -\mathstrut \) \(1010785\) \(x^{5}\mathstrut -\mathstrut \) \(593552\) \(x^{4}\mathstrut +\mathstrut \) \(965980\) \(x^{3}\mathstrut -\mathstrut \) \(88040\) \(x^{2}\mathstrut -\mathstrut \) \(184992\) \(x\mathstrut +\mathstrut \) \(44032\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 6 \)
\(\beta_{3}\)\(=\)\((\)\(144741460206906469\) \(\nu^{17}\mathstrut +\mathstrut \) \(161385737559845937\) \(\nu^{16}\mathstrut -\mathstrut \) \(2664092971879301505\) \(\nu^{15}\mathstrut -\mathstrut \) \(17968626709647954511\) \(\nu^{14}\mathstrut -\mathstrut \) \(87376519471038953432\) \(\nu^{13}\mathstrut +\mathstrut \) \(553399458730912894408\) \(\nu^{12}\mathstrut +\mathstrut \) \(3190063723787691450077\) \(\nu^{11}\mathstrut -\mathstrut \) \(7513705551222347250631\) \(\nu^{10}\mathstrut -\mathstrut \) \(38968899802208855225958\) \(\nu^{9}\mathstrut +\mathstrut \) \(50625426900815744951398\) \(\nu^{8}\mathstrut +\mathstrut \) \(226597310147425552425648\) \(\nu^{7}\mathstrut -\mathstrut \) \(170619435125534118845214\) \(\nu^{6}\mathstrut -\mathstrut \) \(642261520278539445330903\) \(\nu^{5}\mathstrut +\mathstrut \) \(269387139752365671412913\) \(\nu^{4}\mathstrut +\mathstrut \) \(800181860496836468681484\) \(\nu^{3}\mathstrut -\mathstrut \) \(156752961236298346166684\) \(\nu^{2}\mathstrut -\mathstrut \) \(292470729073615043961576\) \(\nu\mathstrut +\mathstrut \) \(10298495692685269437888\)\()/\)\(10\!\cdots\!60\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(46982790863309368\) \(\nu^{17}\mathstrut +\mathstrut \) \(287887730215296643\) \(\nu^{16}\mathstrut +\mathstrut \) \(2529146575510508024\) \(\nu^{15}\mathstrut -\mathstrut \) \(14435549603236349753\) \(\nu^{14}\mathstrut -\mathstrut \) \(58045238973919601552\) \(\nu^{13}\mathstrut +\mathstrut \) \(289263503340686601154\) \(\nu^{12}\mathstrut +\mathstrut \) \(728669155338953426638\) \(\nu^{11}\mathstrut -\mathstrut \) \(2960710970894611138855\) \(\nu^{10}\mathstrut -\mathstrut \) \(5258766929123310518034\) \(\nu^{9}\mathstrut +\mathstrut \) \(16517382496629045041226\) \(\nu^{8}\mathstrut +\mathstrut \) \(20810608458284654194048\) \(\nu^{7}\mathstrut -\mathstrut \) \(50355187934410165785640\) \(\nu^{6}\mathstrut -\mathstrut \) \(39014585577366013427430\) \(\nu^{5}\mathstrut +\mathstrut \) \(80827694791271909898145\) \(\nu^{4}\mathstrut +\mathstrut \) \(24602149696580595655730\) \(\nu^{3}\mathstrut -\mathstrut \) \(57685885791907788408336\) \(\nu^{2}\mathstrut -\mathstrut \) \(536889117329382034640\) \(\nu\mathstrut +\mathstrut \) \(7438077202251186709664\)\()/\)\(10\!\cdots\!16\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(881298394239452567\) \(\nu^{17}\mathstrut -\mathstrut \) \(1278437540638349421\) \(\nu^{16}\mathstrut +\mathstrut \) \(56066780845186860115\) \(\nu^{15}\mathstrut +\mathstrut \) \(64277725368298776203\) \(\nu^{14}\mathstrut -\mathstrut \) \(1437357737846063330984\) \(\nu^{13}\mathstrut -\mathstrut \) \(1253872700329297813884\) \(\nu^{12}\mathstrut +\mathstrut \) \(19178282730478723729149\) \(\nu^{11}\mathstrut +\mathstrut \) \(11675287006380368742663\) \(\nu^{10}\mathstrut -\mathstrut \) \(143082938279936445719626\) \(\nu^{9}\mathstrut -\mathstrut \) \(49064327241059119679454\) \(\nu^{8}\mathstrut +\mathstrut \) \(593520512617502655323496\) \(\nu^{7}\mathstrut +\mathstrut \) \(40706604158487717783202\) \(\nu^{6}\mathstrut -\mathstrut \) \(1264921098665959300354551\) \(\nu^{5}\mathstrut +\mathstrut \) \(267431421777269995760471\) \(\nu^{4}\mathstrut +\mathstrut \) \(1109990961703023245946288\) \(\nu^{3}\mathstrut -\mathstrut \) \(512572453296151056232788\) \(\nu^{2}\mathstrut -\mathstrut \) \(178347090767406941164632\) \(\nu\mathstrut +\mathstrut \) \(83964085877942714478016\)\()/\)\(10\!\cdots\!60\)
\(\beta_{6}\)\(=\)\((\)\(647324479524693901\) \(\nu^{17}\mathstrut +\mathstrut \) \(186381074320600073\) \(\nu^{16}\mathstrut -\mathstrut \) \(37119159518556580905\) \(\nu^{15}\mathstrut -\mathstrut \) \(12649423000801081849\) \(\nu^{14}\mathstrut +\mathstrut \) \(863894573404330912412\) \(\nu^{13}\mathstrut +\mathstrut \) \(292128125616349134102\) \(\nu^{12}\mathstrut -\mathstrut \) \(10521126032523415649447\) \(\nu^{11}\mathstrut -\mathstrut \) \(2897657350783145440659\) \(\nu^{10}\mathstrut +\mathstrut \) \(72245328331576295616598\) \(\nu^{9}\mathstrut +\mathstrut \) \(11039378684800941557632\) \(\nu^{8}\mathstrut -\mathstrut \) \(281074623679790383800628\) \(\nu^{7}\mathstrut +\mathstrut \) \(3853289678616441561834\) \(\nu^{6}\mathstrut +\mathstrut \) \(583621697178849228406593\) \(\nu^{5}\mathstrut -\mathstrut \) \(108118490906802951357763\) \(\nu^{4}\mathstrut -\mathstrut \) \(530567922023584355023904\) \(\nu^{3}\mathstrut +\mathstrut \) \(163850668706739159378334\) \(\nu^{2}\mathstrut +\mathstrut \) \(97542506501936197283516\) \(\nu\mathstrut -\mathstrut \) \(27268050824483756699808\)\()/\)\(27\!\cdots\!40\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(1319417213246488579\) \(\nu^{17}\mathstrut +\mathstrut \) \(559621492421632043\) \(\nu^{16}\mathstrut +\mathstrut \) \(74836300859781871020\) \(\nu^{15}\mathstrut -\mathstrut \) \(21477213972480812264\) \(\nu^{14}\mathstrut -\mathstrut \) \(1731665234056488360618\) \(\nu^{13}\mathstrut +\mathstrut \) \(345047342307579698602\) \(\nu^{12}\mathstrut +\mathstrut \) \(21063274415016879894453\) \(\nu^{11}\mathstrut -\mathstrut \) \(3475333682694168256039\) \(\nu^{10}\mathstrut -\mathstrut \) \(144826605539461435283477\) \(\nu^{9}\mathstrut +\mathstrut \) \(26704603084589679379217\) \(\nu^{8}\mathstrut +\mathstrut \) \(563173170455269203459977\) \(\nu^{7}\mathstrut -\mathstrut \) \(139371686073367841354421\) \(\nu^{6}\mathstrut -\mathstrut \) \(1160593617691666507894842\) \(\nu^{5}\mathstrut +\mathstrut \) \(382409842410729262096402\) \(\nu^{4}\mathstrut +\mathstrut \) \(1049371393018944254993386\) \(\nu^{3}\mathstrut -\mathstrut \) \(404477698360518464795476\) \(\nu^{2}\mathstrut -\mathstrut \) \(225059802638606707352824\) \(\nu\mathstrut +\mathstrut \) \(72283516326198723623712\)\()/\)\(27\!\cdots\!40\)
\(\beta_{8}\)\(=\)\((\)\(2680221458559958891\) \(\nu^{17}\mathstrut +\mathstrut \) \(4145795850755715003\) \(\nu^{16}\mathstrut -\mathstrut \) \(149726000100465944725\) \(\nu^{15}\mathstrut -\mathstrut \) \(235742700843037592569\) \(\nu^{14}\mathstrut +\mathstrut \) \(3339153016499071391482\) \(\nu^{13}\mathstrut +\mathstrut \) \(5161649032350753138572\) \(\nu^{12}\mathstrut -\mathstrut \) \(38130968551015426874657\) \(\nu^{11}\mathstrut -\mathstrut \) \(55029540731354442768569\) \(\nu^{10}\mathstrut +\mathstrut \) \(238904740881430802656228\) \(\nu^{9}\mathstrut +\mathstrut \) \(295990972138672881332042\) \(\nu^{8}\mathstrut -\mathstrut \) \(824662243429869638574348\) \(\nu^{7}\mathstrut -\mathstrut \) \(745441181760796478539546\) \(\nu^{6}\mathstrut +\mathstrut \) \(1497837974473745789432103\) \(\nu^{5}\mathstrut +\mathstrut \) \(662979925977341021768447\) \(\nu^{4}\mathstrut -\mathstrut \) \(1219584920939668650895014\) \(\nu^{3}\mathstrut +\mathstrut \) \(55025050677462704597804\) \(\nu^{2}\mathstrut +\mathstrut \) \(281455097804933887165896\) \(\nu\mathstrut -\mathstrut \) \(48403839672188172335488\)\()/\)\(54\!\cdots\!80\)
\(\beta_{9}\)\(=\)\((\)\(275362841993192751\) \(\nu^{17}\mathstrut -\mathstrut \) \(11452030592941874\) \(\nu^{16}\mathstrut -\mathstrut \) \(16771451536178954874\) \(\nu^{15}\mathstrut +\mathstrut \) \(1070009864819172723\) \(\nu^{14}\mathstrut +\mathstrut \) \(417106420080963868976\) \(\nu^{13}\mathstrut -\mathstrut \) \(46973411178698856458\) \(\nu^{12}\mathstrut -\mathstrut \) \(5463553760020402072751\) \(\nu^{11}\mathstrut +\mathstrut \) \(1013626068034086319130\) \(\nu^{10}\mathstrut +\mathstrut \) \(40509377877914792948981\) \(\nu^{9}\mathstrut -\mathstrut \) \(11196443821297485847857\) \(\nu^{8}\mathstrut -\mathstrut \) \(169423539702812291607437\) \(\nu^{7}\mathstrut +\mathstrut \) \(63133363747566862261845\) \(\nu^{6}\mathstrut +\mathstrut \) \(371026276331576814826644\) \(\nu^{5}\mathstrut -\mathstrut \) \(168412675453334613761487\) \(\nu^{4}\mathstrut -\mathstrut \) \(344795999581976595284218\) \(\nu^{3}\mathstrut +\mathstrut \) \(165779512704691684103418\) \(\nu^{2}\mathstrut +\mathstrut \) \(66601831753044700538392\) \(\nu\mathstrut -\mathstrut \) \(23906363180878424743872\)\()/\)\(54\!\cdots\!08\)
\(\beta_{10}\)\(=\)\((\)\(5825803703253346431\) \(\nu^{17}\mathstrut -\mathstrut \) \(1177100212801470607\) \(\nu^{16}\mathstrut -\mathstrut \) \(328158877652171036575\) \(\nu^{15}\mathstrut +\mathstrut \) \(31395566078394038181\) \(\nu^{14}\mathstrut +\mathstrut \) \(7508465992767470319592\) \(\nu^{13}\mathstrut -\mathstrut \) \(352917596658524263548\) \(\nu^{12}\mathstrut -\mathstrut \) \(89849860499157099278957\) \(\nu^{11}\mathstrut +\mathstrut \) \(5543597877531015384141\) \(\nu^{10}\mathstrut +\mathstrut \) \(604513207116950286951278\) \(\nu^{9}\mathstrut -\mathstrut \) \(85040019779238849726278\) \(\nu^{8}\mathstrut -\mathstrut \) \(2290166480546755013661268\) \(\nu^{7}\mathstrut +\mathstrut \) \(617666632240551842844674\) \(\nu^{6}\mathstrut +\mathstrut \) \(4587404393730349877335083\) \(\nu^{5}\mathstrut -\mathstrut \) \(1882190775513389662567263\) \(\nu^{4}\mathstrut -\mathstrut \) \(4011325985177498170611224\) \(\nu^{3}\mathstrut +\mathstrut \) \(2010199264156004513502124\) \(\nu^{2}\mathstrut +\mathstrut \) \(835083903591907966725896\) \(\nu\mathstrut -\mathstrut \) \(362005719087953560870048\)\()/\)\(10\!\cdots\!60\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(420521664468686129\) \(\nu^{17}\mathstrut +\mathstrut \) \(779514453343670529\) \(\nu^{16}\mathstrut +\mathstrut \) \(23313365080544425399\) \(\nu^{15}\mathstrut -\mathstrut \) \(37495498352020738063\) \(\nu^{14}\mathstrut -\mathstrut \) \(533353235401517776718\) \(\nu^{13}\mathstrut +\mathstrut \) \(733385390450997907920\) \(\nu^{12}\mathstrut +\mathstrut \) \(6497927815602892692027\) \(\nu^{11}\mathstrut -\mathstrut \) \(7543931281612472508327\) \(\nu^{10}\mathstrut -\mathstrut \) \(45351273870523846879168\) \(\nu^{9}\mathstrut +\mathstrut \) \(44081716271204361777594\) \(\nu^{8}\mathstrut +\mathstrut \) \(181041590015686663548156\) \(\nu^{7}\mathstrut -\mathstrut \) \(146706930669148526168558\) \(\nu^{6}\mathstrut -\mathstrut \) \(385696885545280403973261\) \(\nu^{5}\mathstrut +\mathstrut \) \(258805548865262679811493\) \(\nu^{4}\mathstrut +\mathstrut \) \(361933351407689842627290\) \(\nu^{3}\mathstrut -\mathstrut \) \(191424702585629058982116\) \(\nu^{2}\mathstrut -\mathstrut \) \(78881484870860605026780\) \(\nu\mathstrut +\mathstrut \) \(26907587640296053779760\)\()/\)\(54\!\cdots\!08\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(5195850810567831269\) \(\nu^{17}\mathstrut +\mathstrut \) \(1536201918351780013\) \(\nu^{16}\mathstrut +\mathstrut \) \(288640657810594109635\) \(\nu^{15}\mathstrut -\mathstrut \) \(41879421212018532829\) \(\nu^{14}\mathstrut -\mathstrut \) \(6509746865104497911688\) \(\nu^{13}\mathstrut +\mathstrut \) \(327883507809751144792\) \(\nu^{12}\mathstrut +\mathstrut \) \(76709126863757935640263\) \(\nu^{11}\mathstrut -\mathstrut \) \(1444114800575728512859\) \(\nu^{10}\mathstrut -\mathstrut \) \(507613448393773486817252\) \(\nu^{9}\mathstrut +\mathstrut \) \(27418248925441279920192\) \(\nu^{8}\mathstrut +\mathstrut \) \(1890402465298724686491742\) \(\nu^{7}\mathstrut -\mathstrut \) \(275323779236037247962956\) \(\nu^{6}\mathstrut -\mathstrut \) \(3730398457237017637272927\) \(\nu^{5}\mathstrut +\mathstrut \) \(967879913270176393069187\) \(\nu^{4}\mathstrut +\mathstrut \) \(3243219206637238199297696\) \(\nu^{3}\mathstrut -\mathstrut \) \(1090295499626038141092036\) \(\nu^{2}\mathstrut -\mathstrut \) \(678972623141048980263944\) \(\nu\mathstrut +\mathstrut \) \(173505961232324330986672\)\()/\)\(54\!\cdots\!80\)
\(\beta_{13}\)\(=\)\((\)\(11605123817737164599\) \(\nu^{17}\mathstrut -\mathstrut \) \(7280678371989096423\) \(\nu^{16}\mathstrut -\mathstrut \) \(683567934204991962195\) \(\nu^{15}\mathstrut +\mathstrut \) \(360295631299187961409\) \(\nu^{14}\mathstrut +\mathstrut \) \(16513156195318742251568\) \(\nu^{13}\mathstrut -\mathstrut \) \(7675218388245477712372\) \(\nu^{12}\mathstrut -\mathstrut \) \(210908746061052478144373\) \(\nu^{11}\mathstrut +\mathstrut \) \(92717392241152513158109\) \(\nu^{10}\mathstrut +\mathstrut \) \(1530208323295219718755602\) \(\nu^{9}\mathstrut -\mathstrut \) \(684349109926268828755642\) \(\nu^{8}\mathstrut -\mathstrut \) \(6287005702813971781640112\) \(\nu^{7}\mathstrut +\mathstrut \) \(2999048915698040236095846\) \(\nu^{6}\mathstrut +\mathstrut \) \(13612285176504513757943847\) \(\nu^{5}\mathstrut -\mathstrut \) \(6940644528263386969202027\) \(\nu^{4}\mathstrut -\mathstrut \) \(12727970525404889910935616\) \(\nu^{3}\mathstrut +\mathstrut \) \(6459138824142946750958156\) \(\nu^{2}\mathstrut +\mathstrut \) \(2681851923703474081439304\) \(\nu\mathstrut -\mathstrut \) \(1058058203782019517744672\)\()/\)\(10\!\cdots\!60\)
\(\beta_{14}\)\(=\)\((\)\(7339053591108286376\) \(\nu^{17}\mathstrut -\mathstrut \) \(2624304772433765027\) \(\nu^{16}\mathstrut -\mathstrut \) \(410982890438662958100\) \(\nu^{15}\mathstrut +\mathstrut \) \(89304205546366529461\) \(\nu^{14}\mathstrut +\mathstrut \) \(9365328163664075396632\) \(\nu^{13}\mathstrut -\mathstrut \) \(1245745999435260255858\) \(\nu^{12}\mathstrut -\mathstrut \) \(111836406519323740147942\) \(\nu^{11}\mathstrut +\mathstrut \) \(12429267440724046687431\) \(\nu^{10}\mathstrut +\mathstrut \) \(752474096120826661523078\) \(\nu^{9}\mathstrut -\mathstrut \) \(115677967020420557397478\) \(\nu^{8}\mathstrut -\mathstrut \) \(2856338418701901523044188\) \(\nu^{7}\mathstrut +\mathstrut \) \(712706696339483881442964\) \(\nu^{6}\mathstrut +\mathstrut \) \(5736977922223463349170058\) \(\nu^{5}\mathstrut -\mathstrut \) \(2101276257402244035268253\) \(\nu^{4}\mathstrut -\mathstrut \) \(5012250051671683737890314\) \(\nu^{3}\mathstrut +\mathstrut \) \(2255771184842710278109664\) \(\nu^{2}\mathstrut +\mathstrut \) \(974568036369572090260696\) \(\nu\mathstrut -\mathstrut \) \(383540652681053553805248\)\()/\)\(54\!\cdots\!80\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(7953741443864133617\) \(\nu^{17}\mathstrut +\mathstrut \) \(3803940712684142424\) \(\nu^{16}\mathstrut +\mathstrut \) \(453368040610120720965\) \(\nu^{15}\mathstrut -\mathstrut \) \(156798841818442653052\) \(\nu^{14}\mathstrut -\mathstrut \) \(10561666810643101991804\) \(\nu^{13}\mathstrut +\mathstrut \) \(2784319905389981725186\) \(\nu^{12}\mathstrut +\mathstrut \) \(129614735014801394276009\) \(\nu^{11}\mathstrut -\mathstrut \) \(30436051237965383356582\) \(\nu^{10}\mathstrut -\mathstrut \) \(901209997265663540435376\) \(\nu^{9}\mathstrut +\mathstrut \) \(232488443324179272314896\) \(\nu^{8}\mathstrut +\mathstrut \) \(3549613408306537888946556\) \(\nu^{7}\mathstrut -\mathstrut \) \(1138295411913799811081598\) \(\nu^{6}\mathstrut -\mathstrut \) \(7402202429093292483579171\) \(\nu^{5}\mathstrut +\mathstrut \) \(2928183490763908162584206\) \(\nu^{4}\mathstrut +\mathstrut \) \(6713651351725090772166798\) \(\nu^{3}\mathstrut -\mathstrut \) \(2926334498547905458048688\) \(\nu^{2}\mathstrut -\mathstrut \) \(1395852401016791649955152\) \(\nu\mathstrut +\mathstrut \) \(490558310694325359061056\)\()/\)\(54\!\cdots\!80\)
\(\beta_{16}\)\(=\)\((\)\(3517017242975331054\) \(\nu^{17}\mathstrut -\mathstrut \) \(593892311313169743\) \(\nu^{16}\mathstrut -\mathstrut \) \(203964789246663827500\) \(\nu^{15}\mathstrut +\mathstrut \) \(18198600250667013299\) \(\nu^{14}\mathstrut +\mathstrut \) \(4829795900759705272108\) \(\nu^{13}\mathstrut -\mathstrut \) \(290446549791807519422\) \(\nu^{12}\mathstrut -\mathstrut \) \(60205922279305808255348\) \(\nu^{11}\mathstrut +\mathstrut \) \(4990765072018982283819\) \(\nu^{10}\mathstrut +\mathstrut \) \(424980627172688262156972\) \(\nu^{9}\mathstrut -\mathstrut \) \(65428032276378108668312\) \(\nu^{8}\mathstrut -\mathstrut \) \(1698364816772654481764742\) \(\nu^{7}\mathstrut +\mathstrut \) \(439261231222936351171126\) \(\nu^{6}\mathstrut +\mathstrut \) \(3590015700427437101125122\) \(\nu^{5}\mathstrut -\mathstrut \) \(1318966510958154962787587\) \(\nu^{4}\mathstrut -\mathstrut \) \(3294937577077359326796386\) \(\nu^{3}\mathstrut +\mathstrut \) \(1420559813344749429076076\) \(\nu^{2}\mathstrut +\mathstrut \) \(695304949291503236189304\) \(\nu\mathstrut -\mathstrut \) \(246858327195112607012992\)\()/\)\(18\!\cdots\!60\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(11359279336415007443\) \(\nu^{17}\mathstrut +\mathstrut \) \(2151524919216924841\) \(\nu^{16}\mathstrut +\mathstrut \) \(651174043633187701755\) \(\nu^{15}\mathstrut -\mathstrut \) \(60502562770267317013\) \(\nu^{14}\mathstrut -\mathstrut \) \(15220028699798306622306\) \(\nu^{13}\mathstrut +\mathstrut \) \(757350321832368379364\) \(\nu^{12}\mathstrut +\mathstrut \) \(186931225065957739625241\) \(\nu^{11}\mathstrut -\mathstrut \) \(11745145350243322778903\) \(\nu^{10}\mathstrut -\mathstrut \) \(1297579614636252954353614\) \(\nu^{9}\mathstrut +\mathstrut \) \(169017215578411091650744\) \(\nu^{8}\mathstrut +\mathstrut \) \(5092832211855895506522574\) \(\nu^{7}\mathstrut -\mathstrut \) \(1212719058412692058606692\) \(\nu^{6}\mathstrut -\mathstrut \) \(10575497625522344500656189\) \(\nu^{5}\mathstrut +\mathstrut \) \(3762977342104685768185139\) \(\nu^{4}\mathstrut +\mathstrut \) \(9545563548750836060965382\) \(\nu^{3}\mathstrut -\mathstrut \) \(4144770968654097764643512\) \(\nu^{2}\mathstrut -\mathstrut \) \(1956847119506184778360808\) \(\nu\mathstrut +\mathstrut \) \(745643984372214340936464\)\()/\)\(54\!\cdots\!80\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(2\) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(2\) \(\beta_{15}\mathstrut +\mathstrut \) \(2\) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(16\) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(58\)
\(\nu^{5}\)\(=\)\(19\) \(\beta_{17}\mathstrut -\mathstrut \) \(16\) \(\beta_{16}\mathstrut -\mathstrut \) \(18\) \(\beta_{15}\mathstrut +\mathstrut \) \(35\) \(\beta_{14}\mathstrut +\mathstrut \) \(33\) \(\beta_{13}\mathstrut +\mathstrut \) \(3\) \(\beta_{12}\mathstrut +\mathstrut \) \(18\) \(\beta_{11}\mathstrut -\mathstrut \) \(51\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(19\) \(\beta_{8}\mathstrut +\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(16\) \(\beta_{6}\mathstrut +\mathstrut \) \(43\) \(\beta_{5}\mathstrut -\mathstrut \) \(22\) \(\beta_{4}\mathstrut -\mathstrut \) \(19\) \(\beta_{3}\mathstrut +\mathstrut \) \(37\) \(\beta_{2}\mathstrut +\mathstrut \) \(136\) \(\beta_{1}\mathstrut +\mathstrut \) \(20\)
\(\nu^{6}\)\(=\)\(37\) \(\beta_{17}\mathstrut -\mathstrut \) \(39\) \(\beta_{16}\mathstrut -\mathstrut \) \(65\) \(\beta_{15}\mathstrut +\mathstrut \) \(43\) \(\beta_{14}\mathstrut +\mathstrut \) \(8\) \(\beta_{13}\mathstrut +\mathstrut \) \(32\) \(\beta_{12}\mathstrut +\mathstrut \) \(8\) \(\beta_{11}\mathstrut -\mathstrut \) \(34\) \(\beta_{10}\mathstrut +\mathstrut \) \(44\) \(\beta_{9}\mathstrut +\mathstrut \) \(53\) \(\beta_{8}\mathstrut +\mathstrut \) \(28\) \(\beta_{7}\mathstrut +\mathstrut \) \(56\) \(\beta_{6}\mathstrut +\mathstrut \) \(51\) \(\beta_{5}\mathstrut -\mathstrut \) \(82\) \(\beta_{4}\mathstrut -\mathstrut \) \(95\) \(\beta_{3}\mathstrut +\mathstrut \) \(238\) \(\beta_{2}\mathstrut +\mathstrut \) \(94\) \(\beta_{1}\mathstrut +\mathstrut \) \(657\)
\(\nu^{7}\)\(=\)\(283\) \(\beta_{17}\mathstrut -\mathstrut \) \(290\) \(\beta_{16}\mathstrut -\mathstrut \) \(339\) \(\beta_{15}\mathstrut +\mathstrut \) \(521\) \(\beta_{14}\mathstrut +\mathstrut \) \(498\) \(\beta_{13}\mathstrut +\mathstrut \) \(95\) \(\beta_{12}\mathstrut +\mathstrut \) \(284\) \(\beta_{11}\mathstrut -\mathstrut \) \(801\) \(\beta_{10}\mathstrut +\mathstrut \) \(84\) \(\beta_{9}\mathstrut +\mathstrut \) \(347\) \(\beta_{8}\mathstrut +\mathstrut \) \(121\) \(\beta_{7}\mathstrut +\mathstrut \) \(248\) \(\beta_{6}\mathstrut +\mathstrut \) \(802\) \(\beta_{5}\mathstrut -\mathstrut \) \(421\) \(\beta_{4}\mathstrut -\mathstrut \) \(313\) \(\beta_{3}\mathstrut +\mathstrut \) \(620\) \(\beta_{2}\mathstrut +\mathstrut \) \(1830\) \(\beta_{1}\mathstrut +\mathstrut \) \(517\)
\(\nu^{8}\)\(=\)\(572\) \(\beta_{17}\mathstrut -\mathstrut \) \(1026\) \(\beta_{16}\mathstrut -\mathstrut \) \(1547\) \(\beta_{15}\mathstrut +\mathstrut \) \(756\) \(\beta_{14}\mathstrut +\mathstrut \) \(365\) \(\beta_{13}\mathstrut +\mathstrut \) \(680\) \(\beta_{12}\mathstrut +\mathstrut \) \(292\) \(\beta_{11}\mathstrut -\mathstrut \) \(924\) \(\beta_{10}\mathstrut +\mathstrut \) \(818\) \(\beta_{9}\mathstrut +\mathstrut \) \(1160\) \(\beta_{8}\mathstrut +\mathstrut \) \(632\) \(\beta_{7}\mathstrut +\mathstrut \) \(1146\) \(\beta_{6}\mathstrut +\mathstrut \) \(1479\) \(\beta_{5}\mathstrut -\mathstrut \) \(1729\) \(\beta_{4}\mathstrut -\mathstrut \) \(1756\) \(\beta_{3}\mathstrut +\mathstrut \) \(3590\) \(\beta_{2}\mathstrut +\mathstrut \) \(2148\) \(\beta_{1}\mathstrut +\mathstrut \) \(8037\)
\(\nu^{9}\)\(=\)\(3938\) \(\beta_{17}\mathstrut -\mathstrut \) \(5554\) \(\beta_{16}\mathstrut -\mathstrut \) \(6489\) \(\beta_{15}\mathstrut +\mathstrut \) \(7527\) \(\beta_{14}\mathstrut +\mathstrut \) \(7625\) \(\beta_{13}\mathstrut +\mathstrut \) \(2160\) \(\beta_{12}\mathstrut +\mathstrut \) \(4409\) \(\beta_{11}\mathstrut -\mathstrut \) \(12693\) \(\beta_{10}\mathstrut +\mathstrut \) \(2168\) \(\beta_{9}\mathstrut +\mathstrut \) \(6350\) \(\beta_{8}\mathstrut +\mathstrut \) \(1627\) \(\beta_{7}\mathstrut +\mathstrut \) \(3979\) \(\beta_{6}\mathstrut +\mathstrut \) \(14334\) \(\beta_{5}\mathstrut -\mathstrut \) \(7734\) \(\beta_{4}\mathstrut -\mathstrut \) \(5176\) \(\beta_{3}\mathstrut +\mathstrut \) \(10290\) \(\beta_{2}\mathstrut +\mathstrut \) \(26095\) \(\beta_{1}\mathstrut +\mathstrut \) \(9964\)
\(\nu^{10}\)\(=\)\(8515\) \(\beta_{17}\mathstrut -\mathstrut \) \(22545\) \(\beta_{16}\mathstrut -\mathstrut \) \(31986\) \(\beta_{15}\mathstrut +\mathstrut \) \(12702\) \(\beta_{14}\mathstrut +\mathstrut \) \(10037\) \(\beta_{13}\mathstrut +\mathstrut \) \(12850\) \(\beta_{12}\mathstrut +\mathstrut \) \(7185\) \(\beta_{11}\mathstrut -\mathstrut \) \(21127\) \(\beta_{10}\mathstrut +\mathstrut \) \(14610\) \(\beta_{9}\mathstrut +\mathstrut \) \(23263\) \(\beta_{8}\mathstrut +\mathstrut \) \(12495\) \(\beta_{7}\mathstrut +\mathstrut \) \(20895\) \(\beta_{6}\mathstrut +\mathstrut \) \(34011\) \(\beta_{5}\mathstrut -\mathstrut \) \(33002\) \(\beta_{4}\mathstrut -\mathstrut \) \(30057\) \(\beta_{3}\mathstrut +\mathstrut \) \(55328\) \(\beta_{2}\mathstrut +\mathstrut \) \(43110\) \(\beta_{1}\mathstrut +\mathstrut \) \(103555\)
\(\nu^{11}\)\(=\)\(53730\) \(\beta_{17}\mathstrut -\mathstrut \) \(106099\) \(\beta_{16}\mathstrut -\mathstrut \) \(123267\) \(\beta_{15}\mathstrut +\mathstrut \) \(109097\) \(\beta_{14}\mathstrut +\mathstrut \) \(119779\) \(\beta_{13}\mathstrut +\mathstrut \) \(43465\) \(\beta_{12}\mathstrut +\mathstrut \) \(69313\) \(\beta_{11}\mathstrut -\mathstrut \) \(204448\) \(\beta_{10}\mathstrut +\mathstrut \) \(46536\) \(\beta_{9}\mathstrut +\mathstrut \) \(115166\) \(\beta_{8}\mathstrut +\mathstrut \) \(26962\) \(\beta_{7}\mathstrut +\mathstrut \) \(65781\) \(\beta_{6}\mathstrut +\mathstrut \) \(251075\) \(\beta_{5}\mathstrut -\mathstrut \) \(138993\) \(\beta_{4}\mathstrut -\mathstrut \) \(87472\) \(\beta_{3}\mathstrut +\mathstrut \) \(171289\) \(\beta_{2}\mathstrut +\mathstrut \) \(387636\) \(\beta_{1}\mathstrut +\mathstrut \) \(173562\)
\(\nu^{12}\)\(=\)\(125913\) \(\beta_{17}\mathstrut -\mathstrut \) \(450720\) \(\beta_{16}\mathstrut -\mathstrut \) \(613797\) \(\beta_{15}\mathstrut +\mathstrut \) \(211525\) \(\beta_{14}\mathstrut +\mathstrut \) \(224722\) \(\beta_{13}\mathstrut +\mathstrut \) \(232245\) \(\beta_{12}\mathstrut +\mathstrut \) \(150880\) \(\beta_{11}\mathstrut -\mathstrut \) \(434931\) \(\beta_{10}\mathstrut +\mathstrut \) \(257392\) \(\beta_{9}\mathstrut +\mathstrut \) \(442033\) \(\beta_{8}\mathstrut +\mathstrut \) \(228513\) \(\beta_{7}\mathstrut +\mathstrut \) \(361222\) \(\beta_{6}\mathstrut +\mathstrut \) \(696810\) \(\beta_{5}\mathstrut -\mathstrut \) \(599187\) \(\beta_{4}\mathstrut -\mathstrut \) \(500867\) \(\beta_{3}\mathstrut +\mathstrut \) \(869077\) \(\beta_{2}\mathstrut +\mathstrut \) \(809805\) \(\beta_{1}\mathstrut +\mathstrut \) \(1392193\)
\(\nu^{13}\)\(=\)\(733565\) \(\beta_{17}\mathstrut -\mathstrut \) \(1986331\) \(\beta_{16}\mathstrut -\mathstrut \) \(2303444\) \(\beta_{15}\mathstrut +\mathstrut \) \(1604488\) \(\beta_{14}\mathstrut +\mathstrut \) \(1921903\) \(\beta_{13}\mathstrut +\mathstrut \) \(824730\) \(\beta_{12}\mathstrut +\mathstrut \) \(1109565\) \(\beta_{11}\mathstrut -\mathstrut \) \(3334925\) \(\beta_{10}\mathstrut +\mathstrut \) \(911720\) \(\beta_{9}\mathstrut +\mathstrut \) \(2060781\) \(\beta_{8}\mathstrut +\mathstrut \) \(498307\) \(\beta_{7}\mathstrut +\mathstrut \) \(1108007\) \(\beta_{6}\mathstrut +\mathstrut \) \(4344763\) \(\beta_{5}\mathstrut -\mathstrut \) \(2458260\) \(\beta_{4}\mathstrut -\mathstrut \) \(1501553\) \(\beta_{3}\mathstrut +\mathstrut \) \(2862245\) \(\beta_{2}\mathstrut +\mathstrut \) \(5931855\) \(\beta_{1}\mathstrut +\mathstrut \) \(2899559\)
\(\nu^{14}\)\(=\)\(1869123\) \(\beta_{17}\mathstrut -\mathstrut \) \(8530036\) \(\beta_{16}\mathstrut -\mathstrut \) \(11281062\) \(\beta_{15}\mathstrut +\mathstrut \) \(3528593\) \(\beta_{14}\mathstrut +\mathstrut \) \(4532679\) \(\beta_{13}\mathstrut +\mathstrut \) \(4110521\) \(\beta_{12}\mathstrut +\mathstrut \) \(2924390\) \(\beta_{11}\mathstrut -\mathstrut \) \(8400425\) \(\beta_{10}\mathstrut +\mathstrut \) \(4500738\) \(\beta_{9}\mathstrut +\mathstrut \) \(8108143\) \(\beta_{8}\mathstrut +\mathstrut \) \(4000623\) \(\beta_{7}\mathstrut +\mathstrut \) \(6082370\) \(\beta_{6}\mathstrut +\mathstrut \) \(13377401\) \(\beta_{5}\mathstrut -\mathstrut \) \(10577618\) \(\beta_{4}\mathstrut -\mathstrut \) \(8280309\) \(\beta_{3}\mathstrut +\mathstrut \) \(13865394\) \(\beta_{2}\mathstrut +\mathstrut \) \(14651781\) \(\beta_{1}\mathstrut +\mathstrut \) \(19431168\)
\(\nu^{15}\)\(=\)\(10129166\) \(\beta_{17}\mathstrut -\mathstrut \) \(36393606\) \(\beta_{16}\mathstrut -\mathstrut \) \(42279036\) \(\beta_{15}\mathstrut +\mathstrut \) \(24036289\) \(\beta_{14}\mathstrut +\mathstrut \) \(31311162\) \(\beta_{13}\mathstrut +\mathstrut \) \(15135100\) \(\beta_{12}\mathstrut +\mathstrut \) \(18050065\) \(\beta_{11}\mathstrut -\mathstrut \) \(54872037\) \(\beta_{10}\mathstrut +\mathstrut \) \(16964724\) \(\beta_{9}\mathstrut +\mathstrut \) \(36403338\) \(\beta_{8}\mathstrut +\mathstrut \) \(9459343\) \(\beta_{7}\mathstrut +\mathstrut \) \(18845925\) \(\beta_{6}\mathstrut +\mathstrut \) \(74579113\) \(\beta_{5}\mathstrut -\mathstrut \) \(42945283\) \(\beta_{4}\mathstrut -\mathstrut \) \(25966600\) \(\beta_{3}\mathstrut +\mathstrut \) \(47968995\) \(\beta_{2}\mathstrut +\mathstrut \) \(92805501\) \(\beta_{1}\mathstrut +\mathstrut \) \(47620335\)
\(\nu^{16}\)\(=\)\(27986180\) \(\beta_{17}\mathstrut -\mathstrut \) \(156013338\) \(\beta_{16}\mathstrut -\mathstrut \) \(201909588\) \(\beta_{15}\mathstrut +\mathstrut \) \(59111791\) \(\beta_{14}\mathstrut +\mathstrut \) \(86138930\) \(\beta_{13}\mathstrut +\mathstrut \) \(71885464\) \(\beta_{12}\mathstrut +\mathstrut \) \(54152729\) \(\beta_{11}\mathstrut -\mathstrut \) \(155840123\) \(\beta_{10}\mathstrut +\mathstrut \) \(78238866\) \(\beta_{9}\mathstrut +\mathstrut \) \(145244684\) \(\beta_{8}\mathstrut +\mathstrut \) \(68407717\) \(\beta_{7}\mathstrut +\mathstrut \) \(101114245\) \(\beta_{6}\mathstrut +\mathstrut \) \(246814713\) \(\beta_{5}\mathstrut -\mathstrut \) \(183639455\) \(\beta_{4}\mathstrut -\mathstrut \) \(136840878\) \(\beta_{3}\mathstrut +\mathstrut \) \(223987028\) \(\beta_{2}\mathstrut +\mathstrut \) \(259137686\) \(\beta_{1}\mathstrut +\mathstrut \) \(280429505\)
\(\nu^{17}\)\(=\)\(142315607\) \(\beta_{17}\mathstrut -\mathstrut \) \(654644465\) \(\beta_{16}\mathstrut -\mathstrut \) \(763417207\) \(\beta_{15}\mathstrut +\mathstrut \) \(366958297\) \(\beta_{14}\mathstrut +\mathstrut \) \(515499788\) \(\beta_{13}\mathstrut +\mathstrut \) \(271872528\) \(\beta_{12}\mathstrut +\mathstrut \) \(297247588\) \(\beta_{11}\mathstrut -\mathstrut \) \(908224740\) \(\beta_{10}\mathstrut +\mathstrut \) \(305908278\) \(\beta_{9}\mathstrut +\mathstrut \) \(636208503\) \(\beta_{8}\mathstrut +\mathstrut \) \(177988820\) \(\beta_{7}\mathstrut +\mathstrut \) \(321843262\) \(\beta_{6}\mathstrut +\mathstrut \) \(1273093897\) \(\beta_{5}\mathstrut -\mathstrut \) \(743275464\) \(\beta_{4}\mathstrut -\mathstrut \) \(449775207\) \(\beta_{3}\mathstrut +\mathstrut \) \(805581487\) \(\beta_{2}\mathstrut +\mathstrut \) \(1476634875\) \(\beta_{1}\mathstrut +\mathstrut \) \(777857687\)

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
−3.61197
−3.36568
−3.09022
−3.06473
−2.48653
−2.19329
−1.35939
−0.493430
0.301624
0.693705
0.750756
1.51521
1.75663
1.86531
2.64131
3.41379
3.61923
4.10767
0 1.00000 0 −3.61197 0 3.30181 0 1.00000 0
1.2 0 1.00000 0 −3.36568 0 −2.47862 0 1.00000 0
1.3 0 1.00000 0 −3.09022 0 4.20154 0 1.00000 0
1.4 0 1.00000 0 −3.06473 0 −3.46092 0 1.00000 0
1.5 0 1.00000 0 −2.48653 0 0.647451 0 1.00000 0
1.6 0 1.00000 0 −2.19329 0 −1.02399 0 1.00000 0
1.7 0 1.00000 0 −1.35939 0 −0.0125376 0 1.00000 0
1.8 0 1.00000 0 −0.493430 0 −4.29258 0 1.00000 0
1.9 0 1.00000 0 0.301624 0 4.56963 0 1.00000 0
1.10 0 1.00000 0 0.693705 0 −3.39358 0 1.00000 0
1.11 0 1.00000 0 0.750756 0 2.83375 0 1.00000 0
1.12 0 1.00000 0 1.51521 0 0.0590047 0 1.00000 0
1.13 0 1.00000 0 1.75663 0 3.11001 0 1.00000 0
1.14 0 1.00000 0 1.86531 0 −3.57984 0 1.00000 0
1.15 0 1.00000 0 2.64131 0 4.83855 0 1.00000 0
1.16 0 1.00000 0 3.41379 0 −0.715150 0 1.00000 0
1.17 0 1.00000 0 3.61923 0 0.0319200 0 1.00000 0
1.18 0 1.00000 0 4.10767 0 2.36355 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(251\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6024))\):

\(T_{5}^{18} - \cdots\)
\(T_{7}^{18} - \cdots\)