Properties

Label 8024.2.a.w
Level 8024
Weight 2
Character orbit 8024.a
Self dual Yes
Analytic conductor 64.072
Analytic rank 1
Dimension 20
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8024.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0719625819\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{17} q^{5} \) \( + \beta_{15} q^{7} \) \( + \beta_{2} q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{17} q^{5} \) \( + \beta_{15} q^{7} \) \( + \beta_{2} q^{9} \) \( + \beta_{6} q^{11} \) \( + ( -\beta_{3} - \beta_{17} ) q^{13} \) \( + \beta_{7} q^{15} \) \(+ q^{17}\) \( + ( -\beta_{6} + \beta_{8} - \beta_{12} - \beta_{18} ) q^{19} \) \( + ( -2 + \beta_{4} + \beta_{8} + \beta_{9} - \beta_{13} + \beta_{15} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{21} \) \( + ( 1 - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} - \beta_{17} ) q^{23} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{8} + \beta_{13} - \beta_{16} + \beta_{17} ) q^{25} \) \( + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{8} + \beta_{10} + 2 \beta_{12} - 2 \beta_{15} + \beta_{18} + \beta_{19} ) q^{27} \) \( + ( -2 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} - \beta_{15} + \beta_{16} + \beta_{18} + \beta_{19} ) q^{29} \) \( + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{10} + \beta_{12} - \beta_{15} - 2 \beta_{17} + \beta_{18} - \beta_{19} ) q^{31} \) \( + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{8} - \beta_{11} + \beta_{12} + \beta_{17} ) q^{33} \) \( + ( -2 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{14} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{35} \) \( + ( -\beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} - \beta_{14} - \beta_{15} + \beta_{18} ) q^{37} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{11} + 2 \beta_{12} - \beta_{16} + \beta_{17} ) q^{39} \) \( + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{12} - \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{41} \) \( + ( 1 + \beta_{4} + \beta_{7} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{18} - \beta_{19} ) q^{43} \) \( + ( -\beta_{1} - \beta_{4} - \beta_{6} - \beta_{12} - \beta_{17} + \beta_{19} ) q^{45} \) \( + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} + 2 \beta_{7} - \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} - \beta_{16} + 3 \beta_{17} - \beta_{18} + \beta_{19} ) q^{47} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} - \beta_{15} + \beta_{19} ) q^{49} \) \( -\beta_{1} q^{51} \) \( + ( -\beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{15} - 2 \beta_{16} - \beta_{18} ) q^{53} \) \( + ( -2 - 4 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} + 3 \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{16} + \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{55} \) \( + ( -3 - \beta_{2} + \beta_{8} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{18} ) q^{57} \) \(- q^{59}\) \( + ( -1 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{11} + 2 \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{16} - 2 \beta_{18} ) q^{61} \) \( + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{13} + \beta_{14} + \beta_{16} - \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{63} \) \( + ( -1 - \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{16} - 5 \beta_{17} - \beta_{19} ) q^{65} \) \( + ( 1 + 3 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{16} + \beta_{17} - \beta_{19} ) q^{67} \) \( + ( 1 + \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{12} + 2 \beta_{13} - 2 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} + 3 \beta_{18} ) q^{69} \) \( + ( 1 + 2 \beta_{1} + \beta_{3} - 2 \beta_{8} + \beta_{11} + 2 \beta_{12} - \beta_{15} - \beta_{16} - \beta_{17} - \beta_{19} ) q^{71} \) \( + ( -4 + 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{12} - 2 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{73} \) \( + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} - \beta_{17} + \beta_{18} ) q^{75} \) \( + ( 1 + 3 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{8} + 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{77} \) \( + ( 2 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{18} ) q^{79} \) \( + ( 2 + \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - \beta_{14} - 2 \beta_{16} ) q^{81} \) \( + ( -3 + 3 \beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} + 2 \beta_{18} ) q^{83} \) \( + \beta_{17} q^{85} \) \( + ( -2 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{8} - 2 \beta_{10} - 4 \beta_{12} + 2 \beta_{15} + 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{87} \) \( + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + 2 \beta_{17} + \beta_{19} ) q^{89} \) \( + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{17} + \beta_{18} ) q^{91} \) \( + ( 2 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{93} \) \( + ( 2 + 7 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + \beta_{11} + 4 \beta_{12} - \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} - 2 \beta_{19} ) q^{95} \) \( + ( -5 - 2 \beta_{1} + \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{97} \) \( + ( 4 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} + 3 \beta_{8} - 2 \beta_{10} + \beta_{11} - 3 \beta_{12} + \beta_{15} - \beta_{16} - 2 \beta_{18} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(20q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(20q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 8q^{11} \) \(\mathstrut -\mathstrut 3q^{13} \) \(\mathstrut -\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 20q^{17} \) \(\mathstrut -\mathstrut 9q^{19} \) \(\mathstrut -\mathstrut 31q^{21} \) \(\mathstrut -\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 17q^{27} \) \(\mathstrut -\mathstrut 37q^{29} \) \(\mathstrut -\mathstrut 9q^{31} \) \(\mathstrut -\mathstrut 8q^{33} \) \(\mathstrut -\mathstrut 8q^{35} \) \(\mathstrut -\mathstrut 11q^{37} \) \(\mathstrut -\mathstrut 12q^{39} \) \(\mathstrut -\mathstrut 27q^{41} \) \(\mathstrut +\mathstrut 17q^{43} \) \(\mathstrut -\mathstrut 4q^{45} \) \(\mathstrut +\mathstrut 7q^{47} \) \(\mathstrut -\mathstrut 9q^{49} \) \(\mathstrut -\mathstrut 2q^{51} \) \(\mathstrut -\mathstrut q^{53} \) \(\mathstrut -\mathstrut 21q^{55} \) \(\mathstrut -\mathstrut 57q^{57} \) \(\mathstrut -\mathstrut 20q^{59} \) \(\mathstrut -\mathstrut 12q^{61} \) \(\mathstrut -\mathstrut 14q^{63} \) \(\mathstrut -\mathstrut 59q^{65} \) \(\mathstrut -\mathstrut 26q^{67} \) \(\mathstrut +\mathstrut 14q^{69} \) \(\mathstrut -\mathstrut 3q^{71} \) \(\mathstrut -\mathstrut 45q^{73} \) \(\mathstrut +\mathstrut 15q^{75} \) \(\mathstrut +\mathstrut 6q^{77} \) \(\mathstrut +\mathstrut 5q^{79} \) \(\mathstrut +\mathstrut 12q^{81} \) \(\mathstrut -\mathstrut 17q^{83} \) \(\mathstrut -\mathstrut 2q^{85} \) \(\mathstrut -\mathstrut 32q^{87} \) \(\mathstrut -\mathstrut 40q^{89} \) \(\mathstrut -\mathstrut 36q^{91} \) \(\mathstrut -\mathstrut 4q^{93} \) \(\mathstrut +\mathstrut 5q^{95} \) \(\mathstrut -\mathstrut 77q^{97} \) \(\mathstrut +\mathstrut 54q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20}\mathstrut -\mathstrut \) \(2\) \(x^{19}\mathstrut -\mathstrut \) \(29\) \(x^{18}\mathstrut +\mathstrut \) \(51\) \(x^{17}\mathstrut +\mathstrut \) \(341\) \(x^{16}\mathstrut -\mathstrut \) \(514\) \(x^{15}\mathstrut -\mathstrut \) \(2114\) \(x^{14}\mathstrut +\mathstrut \) \(2629\) \(x^{13}\mathstrut +\mathstrut \) \(7532\) \(x^{12}\mathstrut -\mathstrut \) \(7342\) \(x^{11}\mathstrut -\mathstrut \) \(15668\) \(x^{10}\mathstrut +\mathstrut \) \(11260\) \(x^{9}\mathstrut +\mathstrut \) \(18269\) \(x^{8}\mathstrut -\mathstrut \) \(9059\) \(x^{7}\mathstrut -\mathstrut \) \(10680\) \(x^{6}\mathstrut +\mathstrut \) \(3386\) \(x^{5}\mathstrut +\mathstrut \) \(2473\) \(x^{4}\mathstrut -\mathstrut \) \(427\) \(x^{3}\mathstrut -\mathstrut \) \(193\) \(x^{2}\mathstrut +\mathstrut \) \(18\) \(x\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(15008099526873\) \(\nu^{19}\mathstrut +\mathstrut \) \(60125553049071\) \(\nu^{18}\mathstrut -\mathstrut \) \(662031832725318\) \(\nu^{17}\mathstrut -\mathstrut \) \(1693927234801715\) \(\nu^{16}\mathstrut +\mathstrut \) \(10857857034782738\) \(\nu^{15}\mathstrut +\mathstrut \) \(19138039203739608\) \(\nu^{14}\mathstrut -\mathstrut \) \(88763268731180322\) \(\nu^{13}\mathstrut -\mathstrut \) \(113035793829052437\) \(\nu^{12}\mathstrut +\mathstrut \) \(398068393698951039\) \(\nu^{11}\mathstrut +\mathstrut \) \(382733090508826625\) \(\nu^{10}\mathstrut -\mathstrut \) \(1003553197024123219\) \(\nu^{9}\mathstrut -\mathstrut \) \(757325016683489087\) \(\nu^{8}\mathstrut +\mathstrut \) \(1390326752397642422\) \(\nu^{7}\mathstrut +\mathstrut \) \(832500114255853691\) \(\nu^{6}\mathstrut -\mathstrut \) \(975831262333912997\) \(\nu^{5}\mathstrut -\mathstrut \) \(439383939498289035\) \(\nu^{4}\mathstrut +\mathstrut \) \(284652088557900286\) \(\nu^{3}\mathstrut +\mathstrut \) \(77862311219996419\) \(\nu^{2}\mathstrut -\mathstrut \) \(19375772586548854\) \(\nu\mathstrut -\mathstrut \) \(2838889908921188\)\()/\)\(324854300335192\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(12145529660002\) \(\nu^{19}\mathstrut +\mathstrut \) \(15707606525101\) \(\nu^{18}\mathstrut +\mathstrut \) \(366647734887630\) \(\nu^{17}\mathstrut -\mathstrut \) \(361220750544340\) \(\nu^{16}\mathstrut -\mathstrut \) \(4513408121091215\) \(\nu^{15}\mathstrut +\mathstrut \) \(3086727461685201\) \(\nu^{14}\mathstrut +\mathstrut \) \(29480854261259565\) \(\nu^{13}\mathstrut -\mathstrut \) \(11590228934722691\) \(\nu^{12}\mathstrut -\mathstrut \) \(111301235840544898\) \(\nu^{11}\mathstrut +\mathstrut \) \(14094405788903625\) \(\nu^{10}\mathstrut +\mathstrut \) \(246389210490892014\) \(\nu^{9}\mathstrut +\mathstrut \) \(24089902119823135\) \(\nu^{8}\mathstrut -\mathstrut \) \(307028737628062186\) \(\nu^{7}\mathstrut -\mathstrut \) \(82227210328563488\) \(\nu^{6}\mathstrut +\mathstrut \) \(193148281561856521\) \(\nu^{5}\mathstrut +\mathstrut \) \(73537393675682740\) \(\nu^{4}\mathstrut -\mathstrut \) \(47892053345017289\) \(\nu^{3}\mathstrut -\mathstrut \) \(21080340326808055\) \(\nu^{2}\mathstrut +\mathstrut \) \(2541759662072204\) \(\nu\mathstrut +\mathstrut \) \(1211960300269868\)\()/\)\(162427150167596\)
\(\beta_{5}\)\(=\)\((\)\(16020819195294\) \(\nu^{19}\mathstrut -\mathstrut \) \(58870048375111\) \(\nu^{18}\mathstrut -\mathstrut \) \(398443278955476\) \(\nu^{17}\mathstrut +\mathstrut \) \(1558574906472928\) \(\nu^{16}\mathstrut +\mathstrut \) \(3770351597264549\) \(\nu^{15}\mathstrut -\mathstrut \) \(16454188961202605\) \(\nu^{14}\mathstrut -\mathstrut \) \(16797496500552161\) \(\nu^{13}\mathstrut +\mathstrut \) \(89562392021593723\) \(\nu^{12}\mathstrut +\mathstrut \) \(33678474302062024\) \(\nu^{11}\mathstrut -\mathstrut \) \(273283568562233103\) \(\nu^{10}\mathstrut -\mathstrut \) \(10812584908264084\) \(\nu^{9}\mathstrut +\mathstrut \) \(476230862128798891\) \(\nu^{8}\mathstrut -\mathstrut \) \(65348280333808960\) \(\nu^{7}\mathstrut -\mathstrut \) \(457770153580060112\) \(\nu^{6}\mathstrut +\mathstrut \) \(96373519748605197\) \(\nu^{5}\mathstrut +\mathstrut \) \(216042989594827158\) \(\nu^{4}\mathstrut -\mathstrut \) \(41788977255725345\) \(\nu^{3}\mathstrut -\mathstrut \) \(36097994348251449\) \(\nu^{2}\mathstrut +\mathstrut \) \(2060400142794990\) \(\nu\mathstrut +\mathstrut \) \(992821189787352\)\()/\)\(162427150167596\)
\(\beta_{6}\)\(=\)\((\)\(76645812152769\) \(\nu^{19}\mathstrut -\mathstrut \) \(17936170631519\) \(\nu^{18}\mathstrut -\mathstrut \) \(2510754831620630\) \(\nu^{17}\mathstrut +\mathstrut \) \(56976029903065\) \(\nu^{16}\mathstrut +\mathstrut \) \(33419691122965832\) \(\nu^{15}\mathstrut +\mathstrut \) \(4892237529723266\) \(\nu^{14}\mathstrut -\mathstrut \) \(234425038995395728\) \(\nu^{13}\mathstrut -\mathstrut \) \(65991058903842495\) \(\nu^{12}\mathstrut +\mathstrut \) \(939297843063741795\) \(\nu^{11}\mathstrut +\mathstrut \) \(363131113987540267\) \(\nu^{10}\mathstrut -\mathstrut \) \(2174981033621691055\) \(\nu^{9}\mathstrut -\mathstrut \) \(1002870192289540545\) \(\nu^{8}\mathstrut +\mathstrut \) \(2802119763452223462\) \(\nu^{7}\mathstrut +\mathstrut \) \(1399019659984449431\) \(\nu^{6}\mathstrut -\mathstrut \) \(1819299238556713483\) \(\nu^{5}\mathstrut -\mathstrut \) \(890798728393047619\) \(\nu^{4}\mathstrut +\mathstrut \) \(475471083274295272\) \(\nu^{3}\mathstrut +\mathstrut \) \(194498856062150625\) \(\nu^{2}\mathstrut -\mathstrut \) \(26389126429491226\) \(\nu\mathstrut -\mathstrut \) \(8688382894365628\)\()/\)\(649708600670384\)
\(\beta_{7}\)\(=\)\((\)\(133807711367713\) \(\nu^{19}\mathstrut -\mathstrut \) \(362894116240403\) \(\nu^{18}\mathstrut -\mathstrut \) \(3518580791424854\) \(\nu^{17}\mathstrut +\mathstrut \) \(9009384589595321\) \(\nu^{16}\mathstrut +\mathstrut \) \(36596855043647524\) \(\nu^{15}\mathstrut -\mathstrut \) \(86824555481956378\) \(\nu^{14}\mathstrut -\mathstrut \) \(195327724726328228\) \(\nu^{13}\mathstrut +\mathstrut \) \(412850241478513821\) \(\nu^{12}\mathstrut +\mathstrut \) \(587750038012930235\) \(\nu^{11}\mathstrut -\mathstrut \) \(1023878782469119281\) \(\nu^{10}\mathstrut -\mathstrut \) \(1032687384505394863\) \(\nu^{9}\mathstrut +\mathstrut \) \(1278742982669404531\) \(\nu^{8}\mathstrut +\mathstrut \) \(1041146529330288966\) \(\nu^{7}\mathstrut -\mathstrut \) \(672256097331311529\) \(\nu^{6}\mathstrut -\mathstrut \) \(557561862694552919\) \(\nu^{5}\mathstrut +\mathstrut \) \(50044312233059173\) \(\nu^{4}\mathstrut +\mathstrut \) \(135097212817084372\) \(\nu^{3}\mathstrut +\mathstrut \) \(23863906306901029\) \(\nu^{2}\mathstrut -\mathstrut \) \(7871426233683754\) \(\nu\mathstrut -\mathstrut \) \(1373594821745228\)\()/\)\(649708600670384\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(42993313897265\) \(\nu^{19}\mathstrut +\mathstrut \) \(67952978646472\) \(\nu^{18}\mathstrut +\mathstrut \) \(1310195314921200\) \(\nu^{17}\mathstrut -\mathstrut \) \(1775850676626261\) \(\nu^{16}\mathstrut -\mathstrut \) \(16202416085915791\) \(\nu^{15}\mathstrut +\mathstrut \) \(18600077304590315\) \(\nu^{14}\mathstrut +\mathstrut \) \(105230105835551111\) \(\nu^{13}\mathstrut -\mathstrut \) \(100594395526169050\) \(\nu^{12}\mathstrut -\mathstrut \) \(387930886724892339\) \(\nu^{11}\mathstrut +\mathstrut \) \(302875427718787716\) \(\nu^{10}\mathstrut +\mathstrut \) \(815185399315337013\) \(\nu^{9}\mathstrut -\mathstrut \) \(511492737522328872\) \(\nu^{8}\mathstrut -\mathstrut \) \(920844546752854704\) \(\nu^{7}\mathstrut +\mathstrut \) \(463145176045539225\) \(\nu^{6}\mathstrut +\mathstrut \) \(474359037002118578\) \(\nu^{5}\mathstrut -\mathstrut \) \(193714951616106737\) \(\nu^{4}\mathstrut -\mathstrut \) \(66401913895826783\) \(\nu^{3}\mathstrut +\mathstrut \) \(20729954385889876\) \(\nu^{2}\mathstrut +\mathstrut \) \(926188356680900\) \(\nu\mathstrut -\mathstrut \) \(219488687815120\)\()/\)\(162427150167596\)
\(\beta_{9}\)\(=\)\((\)\(45331988284303\) \(\nu^{19}\mathstrut -\mathstrut \) \(66818051107591\) \(\nu^{18}\mathstrut -\mathstrut \) \(1356148938261212\) \(\nu^{17}\mathstrut +\mathstrut \) \(1610169485420061\) \(\nu^{16}\mathstrut +\mathstrut \) \(16496086110920408\) \(\nu^{15}\mathstrut -\mathstrut \) \(14923641765964600\) \(\nu^{14}\mathstrut -\mathstrut \) \(106012809629079580\) \(\nu^{13}\mathstrut +\mathstrut \) \(66498123155735181\) \(\nu^{12}\mathstrut +\mathstrut \) \(391495676868724363\) \(\nu^{11}\mathstrut -\mathstrut \) \(143049604633506475\) \(\nu^{10}\mathstrut -\mathstrut \) \(841687619283022289\) \(\nu^{9}\mathstrut +\mathstrut \) \(113991057443289155\) \(\nu^{8}\mathstrut +\mathstrut \) \(1010287773076944792\) \(\nu^{7}\mathstrut +\mathstrut \) \(48018581620782459\) \(\nu^{6}\mathstrut -\mathstrut \) \(606765074011869397\) \(\nu^{5}\mathstrut -\mathstrut \) \(105896011211267101\) \(\nu^{4}\mathstrut +\mathstrut \) \(142773250581423458\) \(\nu^{3}\mathstrut +\mathstrut \) \(33438588650521935\) \(\nu^{2}\mathstrut -\mathstrut \) \(7266793329230786\) \(\nu\mathstrut -\mathstrut \) \(1318382064499428\)\()/\)\(162427150167596\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(184101995107787\) \(\nu^{19}\mathstrut +\mathstrut \) \(343669186525913\) \(\nu^{18}\mathstrut +\mathstrut \) \(5444741689527450\) \(\nu^{17}\mathstrut -\mathstrut \) \(8885338469872435\) \(\nu^{16}\mathstrut -\mathstrut \) \(65349816905214356\) \(\nu^{15}\mathstrut +\mathstrut \) \(91424611791938302\) \(\nu^{14}\mathstrut +\mathstrut \) \(413017817795868572\) \(\nu^{13}\mathstrut -\mathstrut \) \(481851404628608415\) \(\nu^{12}\mathstrut -\mathstrut \) \(1492102673133149905\) \(\nu^{11}\mathstrut +\mathstrut \) \(1402612592468224371\) \(\nu^{10}\mathstrut +\mathstrut \) \(3111824616481401453\) \(\nu^{9}\mathstrut -\mathstrut \) \(2271774499757568745\) \(\nu^{8}\mathstrut -\mathstrut \) \(3564532060563709578\) \(\nu^{7}\mathstrut +\mathstrut \) \(1952850981557902643\) \(\nu^{6}\mathstrut +\mathstrut \) \(1959071038543433605\) \(\nu^{5}\mathstrut -\mathstrut \) \(769321476487400719\) \(\nu^{4}\mathstrut -\mathstrut \) \(372178521461350052\) \(\nu^{3}\mathstrut +\mathstrut \) \(81599324284293657\) \(\nu^{2}\mathstrut +\mathstrut \) \(20728924604707286\) \(\nu\mathstrut -\mathstrut \) \(1533584563137276\)\()/\)\(649708600670384\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(120925478470383\) \(\nu^{19}\mathstrut +\mathstrut \) \(250843014352425\) \(\nu^{18}\mathstrut +\mathstrut \) \(3488891386882426\) \(\nu^{17}\mathstrut -\mathstrut \) \(6434602754611047\) \(\nu^{16}\mathstrut -\mathstrut \) \(40760274008323080\) \(\nu^{15}\mathstrut +\mathstrut \) \(65395288982125354\) \(\nu^{14}\mathstrut +\mathstrut \) \(250563594140693376\) \(\nu^{13}\mathstrut -\mathstrut \) \(338645504851607879\) \(\nu^{12}\mathstrut -\mathstrut \) \(882469490014203597\) \(\nu^{11}\mathstrut +\mathstrut \) \(963571764254885067\) \(\nu^{10}\mathstrut +\mathstrut \) \(1804039893884647513\) \(\nu^{9}\mathstrut -\mathstrut \) \(1518770889869864953\) \(\nu^{8}\mathstrut -\mathstrut \) \(2039264779030142170\) \(\nu^{7}\mathstrut +\mathstrut \) \(1264141963182752687\) \(\nu^{6}\mathstrut +\mathstrut \) \(1110591771138454285\) \(\nu^{5}\mathstrut -\mathstrut \) \(477978667829286251\) \(\nu^{4}\mathstrut -\mathstrut \) \(202750840008521224\) \(\nu^{3}\mathstrut +\mathstrut \) \(46572047820776361\) \(\nu^{2}\mathstrut +\mathstrut \) \(6169267865881726\) \(\nu\mathstrut -\mathstrut \) \(435029148696828\)\()/\)\(324854300335192\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(263860853334045\) \(\nu^{19}\mathstrut +\mathstrut \) \(343389161409139\) \(\nu^{18}\mathstrut +\mathstrut \) \(8117295406965246\) \(\nu^{17}\mathstrut -\mathstrut \) \(8485562766894213\) \(\nu^{16}\mathstrut -\mathstrut \) \(101591826901346296\) \(\nu^{15}\mathstrut +\mathstrut \) \(82161373952496470\) \(\nu^{14}\mathstrut +\mathstrut \) \(670799251315472128\) \(\nu^{13}\mathstrut -\mathstrut \) \(395779143570416637\) \(\nu^{12}\mathstrut -\mathstrut \) \(2533427590015832951\) \(\nu^{11}\mathstrut +\mathstrut \) \(994281512735783585\) \(\nu^{10}\mathstrut +\mathstrut \) \(5527004308621554643\) \(\nu^{9}\mathstrut -\mathstrut \) \(1230247495177683859\) \(\nu^{8}\mathstrut -\mathstrut \) \(6657626202798049406\) \(\nu^{7}\mathstrut +\mathstrut \) \(576773204913683829\) \(\nu^{6}\mathstrut +\mathstrut \) \(3927555504213749439\) \(\nu^{5}\mathstrut +\mathstrut \) \(63860054656180327\) \(\nu^{4}\mathstrut -\mathstrut \) \(852273689684712872\) \(\nu^{3}\mathstrut -\mathstrut \) \(89126045129340541\) \(\nu^{2}\mathstrut +\mathstrut \) \(38672885783596418\) \(\nu\mathstrut +\mathstrut \) \(5087224973914476\)\()/\)\(649708600670384\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(151331030049999\) \(\nu^{19}\mathstrut +\mathstrut \) \(328605339348677\) \(\nu^{18}\mathstrut +\mathstrut \) \(4297621691911310\) \(\nu^{17}\mathstrut -\mathstrut \) \(8352082277543607\) \(\nu^{16}\mathstrut -\mathstrut \) \(49280507797760916\) \(\nu^{15}\mathstrut +\mathstrut \) \(83656310171524474\) \(\nu^{14}\mathstrut +\mathstrut \) \(296597362424934872\) \(\nu^{13}\mathstrut -\mathstrut \) \(423436485953518199\) \(\nu^{12}\mathstrut -\mathstrut \) \(1021744885212321625\) \(\nu^{11}\mathstrut +\mathstrut \) \(1162725972686421455\) \(\nu^{10}\mathstrut +\mathstrut \) \(2045658820045451989\) \(\nu^{9}\mathstrut -\mathstrut \) \(1732814191116594453\) \(\nu^{8}\mathstrut -\mathstrut \) \(2272630270692215286\) \(\nu^{7}\mathstrut +\mathstrut \) \(1316521826657137191\) \(\nu^{6}\mathstrut +\mathstrut \) \(1228023278495875625\) \(\nu^{5}\mathstrut -\mathstrut \) \(423467513867776255\) \(\nu^{4}\mathstrut -\mathstrut \) \(234667880139975948\) \(\nu^{3}\mathstrut +\mathstrut \) \(23651666818882825\) \(\nu^{2}\mathstrut +\mathstrut \) \(11555573337297482\) \(\nu\mathstrut +\mathstrut \) \(1127675225727556\)\()/\)\(324854300335192\)
\(\beta_{14}\)\(=\)\((\)\(80461065143945\) \(\nu^{19}\mathstrut -\mathstrut \) \(192949961716001\) \(\nu^{18}\mathstrut -\mathstrut \) \(2225832307186190\) \(\nu^{17}\mathstrut +\mathstrut \) \(4885900865202215\) \(\nu^{16}\mathstrut +\mathstrut \) \(24737934398925096\) \(\nu^{15}\mathstrut -\mathstrut \) \(48603045690399934\) \(\nu^{14}\mathstrut -\mathstrut \) \(143674251387053078\) \(\nu^{13}\mathstrut +\mathstrut \) \(243289363599210233\) \(\nu^{12}\mathstrut +\mathstrut \) \(477161979795830635\) \(\nu^{11}\mathstrut -\mathstrut \) \(657051372951100069\) \(\nu^{10}\mathstrut -\mathstrut \) \(925368179657787681\) \(\nu^{9}\mathstrut +\mathstrut \) \(955721189306466233\) \(\nu^{8}\mathstrut +\mathstrut \) \(1006134369947103434\) \(\nu^{7}\mathstrut -\mathstrut \) \(700125008525691579\) \(\nu^{6}\mathstrut -\mathstrut \) \(541340773534791955\) \(\nu^{5}\mathstrut +\mathstrut \) \(216446665902964519\) \(\nu^{4}\mathstrut +\mathstrut \) \(105786552896986378\) \(\nu^{3}\mathstrut -\mathstrut \) \(17976868945581973\) \(\nu^{2}\mathstrut -\mathstrut \) \(3551592323591632\) \(\nu\mathstrut +\mathstrut \) \(292932332005348\)\()/\)\(162427150167596\)
\(\beta_{15}\)\(=\)\((\)\(325730164389989\) \(\nu^{19}\mathstrut -\mathstrut \) \(530619240509251\) \(\nu^{18}\mathstrut -\mathstrut \) \(9676518745643606\) \(\nu^{17}\mathstrut +\mathstrut \) \(13102163715311677\) \(\nu^{16}\mathstrut +\mathstrut \) \(116856652484531984\) \(\nu^{15}\mathstrut -\mathstrut \) \(126079224026400470\) \(\nu^{14}\mathstrut -\mathstrut \) \(745542319166916016\) \(\nu^{13}\mathstrut +\mathstrut \) \(599534902485367861\) \(\nu^{12}\mathstrut +\mathstrut \) \(2734315745423201031\) \(\nu^{11}\mathstrut -\mathstrut \) \(1474980530814644993\) \(\nu^{10}\mathstrut -\mathstrut \) \(5842039110756528915\) \(\nu^{9}\mathstrut +\mathstrut \) \(1761207932731377075\) \(\nu^{8}\mathstrut +\mathstrut \) \(6965910068684820806\) \(\nu^{7}\mathstrut -\mathstrut \) \(741110452971478765\) \(\nu^{6}\mathstrut -\mathstrut \) \(4130225627329007871\) \(\nu^{5}\mathstrut -\mathstrut \) \(156624587821843463\) \(\nu^{4}\mathstrut +\mathstrut \) \(935900795295101504\) \(\nu^{3}\mathstrut +\mathstrut \) \(122692605606049573\) \(\nu^{2}\mathstrut -\mathstrut \) \(50663249088470778\) \(\nu\mathstrut -\mathstrut \) \(5804452142803164\)\()/\)\(649708600670384\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(337802121038101\) \(\nu^{19}\mathstrut +\mathstrut \) \(685783496165623\) \(\nu^{18}\mathstrut +\mathstrut \) \(9722347482910126\) \(\nu^{17}\mathstrut -\mathstrut \) \(17370734467795253\) \(\nu^{16}\mathstrut -\mathstrut \) \(113259303600444996\) \(\nu^{15}\mathstrut +\mathstrut \) \(173221349864172914\) \(\nu^{14}\mathstrut +\mathstrut \) \(694218163273023404\) \(\nu^{13}\mathstrut -\mathstrut \) \(870857250127774401\) \(\nu^{12}\mathstrut -\mathstrut \) \(2440377683595008103\) \(\nu^{11}\mathstrut +\mathstrut \) \(2362942329726706325\) \(\nu^{10}\mathstrut +\mathstrut \) \(4995360138329745683\) \(\nu^{9}\mathstrut -\mathstrut \) \(3444621996449845175\) \(\nu^{8}\mathstrut -\mathstrut \) \(5701979413137081982\) \(\nu^{7}\mathstrut +\mathstrut \) \(2514828501259415125\) \(\nu^{6}\mathstrut +\mathstrut \) \(3217787591851186899\) \(\nu^{5}\mathstrut -\mathstrut \) \(756407891702026433\) \(\nu^{4}\mathstrut -\mathstrut \) \(680181069548745356\) \(\nu^{3}\mathstrut +\mathstrut \) \(40764924466738319\) \(\nu^{2}\mathstrut +\mathstrut \) \(36440601667275146\) \(\nu\mathstrut +\mathstrut \) \(1046948189106380\)\()/\)\(649708600670384\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(343398705436307\) \(\nu^{19}\mathstrut +\mathstrut \) \(552989699504901\) \(\nu^{18}\mathstrut +\mathstrut \) \(10321456573893306\) \(\nu^{17}\mathstrut -\mathstrut \) \(13994753185826803\) \(\nu^{16}\mathstrut -\mathstrut \) \(126108343143376008\) \(\nu^{15}\mathstrut +\mathstrut \) \(139910079550614274\) \(\nu^{14}\mathstrut +\mathstrut \) \(812769418774309376\) \(\nu^{13}\mathstrut -\mathstrut \) \(707467471865722875\) \(\nu^{12}\mathstrut -\mathstrut \) \(2999329290824778145\) \(\nu^{11}\mathstrut +\mathstrut \) \(1933483257300435759\) \(\nu^{10}\mathstrut +\mathstrut \) \(6404249699245177357\) \(\nu^{9}\mathstrut -\mathstrut \) \(2833982038707421957\) \(\nu^{8}\mathstrut -\mathstrut \) \(7552293932285297114\) \(\nu^{7}\mathstrut +\mathstrut \) \(2069702343217216147\) \(\nu^{6}\mathstrut +\mathstrut \) \(4339754271391070289\) \(\nu^{5}\mathstrut -\mathstrut \) \(605186153912782583\) \(\nu^{4}\mathstrut -\mathstrut \) \(899269310777046384\) \(\nu^{3}\mathstrut +\mathstrut \) \(11534034404218717\) \(\nu^{2}\mathstrut +\mathstrut \) \(42412043842306222\) \(\nu\mathstrut +\mathstrut \) \(1690249535830228\)\()/\)\(649708600670384\)
\(\beta_{18}\)\(=\)\((\)\(253562826274429\) \(\nu^{19}\mathstrut -\mathstrut \) \(412127320209451\) \(\nu^{18}\mathstrut -\mathstrut \) \(7532771991506694\) \(\nu^{17}\mathstrut +\mathstrut \) \(10164264475770781\) \(\nu^{16}\mathstrut +\mathstrut \) \(90977414723089360\) \(\nu^{15}\mathstrut -\mathstrut \) \(97618304344007118\) \(\nu^{14}\mathstrut -\mathstrut \) \(580606292069891768\) \(\nu^{13}\mathstrut +\mathstrut \) \(462576009374162421\) \(\nu^{12}\mathstrut +\mathstrut \) \(2130966451539884599\) \(\nu^{11}\mathstrut -\mathstrut \) \(1129872805579749937\) \(\nu^{10}\mathstrut -\mathstrut \) \(4560792386051901571\) \(\nu^{9}\mathstrut +\mathstrut \) \(1323862194602930747\) \(\nu^{8}\mathstrut +\mathstrut \) \(5460871544306186342\) \(\nu^{7}\mathstrut -\mathstrut \) \(507756287116343477\) \(\nu^{6}\mathstrut -\mathstrut \) \(3273486924765045663\) \(\nu^{5}\mathstrut -\mathstrut \) \(175910025104156383\) \(\nu^{4}\mathstrut +\mathstrut \) \(765517191406150336\) \(\nu^{3}\mathstrut +\mathstrut \) \(114190401142210877\) \(\nu^{2}\mathstrut -\mathstrut \) \(42553984246960834\) \(\nu\mathstrut -\mathstrut \) \(6028723425803332\)\()/\)\(324854300335192\)
\(\beta_{19}\)\(=\)\((\)\(546278944556307\) \(\nu^{19}\mathstrut -\mathstrut \) \(1079538820370601\) \(\nu^{18}\mathstrut -\mathstrut \) \(15754972942733434\) \(\nu^{17}\mathstrut +\mathstrut \) \(27147611165227707\) \(\nu^{16}\mathstrut +\mathstrut \) \(184038309862958588\) \(\nu^{15}\mathstrut -\mathstrut \) \(267961414877285062\) \(\nu^{14}\mathstrut -\mathstrut \) \(1132310976476300436\) \(\nu^{13}\mathstrut +\mathstrut \) \(1326893469526979935\) \(\nu^{12}\mathstrut +\mathstrut \) \(4000786456942655977\) \(\nu^{11}\mathstrut -\mathstrut \) \(3516988101925865587\) \(\nu^{10}\mathstrut -\mathstrut \) \(8242987434195879221\) \(\nu^{9}\mathstrut +\mathstrut \) \(4934592548073371881\) \(\nu^{8}\mathstrut +\mathstrut \) \(9481058514763719002\) \(\nu^{7}\mathstrut -\mathstrut \) \(3364025157563174907\) \(\nu^{6}\mathstrut -\mathstrut \) \(5395630552419661493\) \(\nu^{5}\mathstrut +\mathstrut \) \(873392466186698711\) \(\nu^{4}\mathstrut +\mathstrut \) \(1149168002702691532\) \(\nu^{3}\mathstrut -\mathstrut \) \(19970081243144425\) \(\nu^{2}\mathstrut -\mathstrut \) \(57941447455183990\) \(\nu\mathstrut -\mathstrut \) \(1598584945561284\)\()/\)\(649708600670384\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{19}\mathstrut -\mathstrut \) \(\beta_{18}\mathstrut +\mathstrut \) \(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(2\) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(2\) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(20\)
\(\nu^{5}\)\(=\)\(-\)\(12\) \(\beta_{19}\mathstrut -\mathstrut \) \(16\) \(\beta_{18}\mathstrut -\mathstrut \) \(2\) \(\beta_{17}\mathstrut -\mathstrut \) \(4\) \(\beta_{16}\mathstrut +\mathstrut \) \(22\) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(23\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(12\) \(\beta_{10}\mathstrut +\mathstrut \) \(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(24\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(8\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(16\)
\(\nu^{6}\)\(=\)\(-\)\(4\) \(\beta_{19}\mathstrut -\mathstrut \) \(14\) \(\beta_{18}\mathstrut +\mathstrut \) \(2\) \(\beta_{17}\mathstrut -\mathstrut \) \(35\) \(\beta_{16}\mathstrut +\mathstrut \) \(7\) \(\beta_{15}\mathstrut -\mathstrut \) \(12\) \(\beta_{14}\mathstrut +\mathstrut \) \(21\) \(\beta_{13}\mathstrut -\mathstrut \) \(38\) \(\beta_{12}\mathstrut +\mathstrut \) \(16\) \(\beta_{11}\mathstrut -\mathstrut \) \(20\) \(\beta_{10}\mathstrut +\mathstrut \) \(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(23\) \(\beta_{8}\mathstrut +\mathstrut \) \(11\) \(\beta_{7}\mathstrut -\mathstrut \) \(23\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(31\) \(\beta_{4}\mathstrut -\mathstrut \) \(28\) \(\beta_{3}\mathstrut +\mathstrut \) \(84\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(168\)
\(\nu^{7}\)\(=\)\(-\)\(119\) \(\beta_{19}\mathstrut -\mathstrut \) \(195\) \(\beta_{18}\mathstrut -\mathstrut \) \(21\) \(\beta_{17}\mathstrut -\mathstrut \) \(80\) \(\beta_{16}\mathstrut +\mathstrut \) \(217\) \(\beta_{15}\mathstrut +\mathstrut \) \(10\) \(\beta_{14}\mathstrut -\mathstrut \) \(28\) \(\beta_{13}\mathstrut -\mathstrut \) \(249\) \(\beta_{12}\mathstrut +\mathstrut \) \(42\) \(\beta_{11}\mathstrut -\mathstrut \) \(135\) \(\beta_{10}\mathstrut +\mathstrut \) \(63\) \(\beta_{9}\mathstrut +\mathstrut \) \(254\) \(\beta_{8}\mathstrut -\mathstrut \) \(16\) \(\beta_{7}\mathstrut -\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(50\) \(\beta_{5}\mathstrut -\mathstrut \) \(47\) \(\beta_{4}\mathstrut -\mathstrut \) \(117\) \(\beta_{3}\mathstrut +\mathstrut \) \(173\) \(\beta_{2}\mathstrut +\mathstrut \) \(102\) \(\beta_{1}\mathstrut +\mathstrut \) \(219\)
\(\nu^{8}\)\(=\)\(-\)\(84\) \(\beta_{19}\mathstrut -\mathstrut \) \(296\) \(\beta_{18}\mathstrut +\mathstrut \) \(39\) \(\beta_{17}\mathstrut -\mathstrut \) \(454\) \(\beta_{16}\mathstrut +\mathstrut \) \(164\) \(\beta_{15}\mathstrut -\mathstrut \) \(113\) \(\beta_{14}\mathstrut +\mathstrut \) \(174\) \(\beta_{13}\mathstrut -\mathstrut \) \(550\) \(\beta_{12}\mathstrut +\mathstrut \) \(215\) \(\beta_{11}\mathstrut -\mathstrut \) \(295\) \(\beta_{10}\mathstrut +\mathstrut \) \(90\) \(\beta_{9}\mathstrut +\mathstrut \) \(372\) \(\beta_{8}\mathstrut +\mathstrut \) \(83\) \(\beta_{7}\mathstrut -\mathstrut \) \(233\) \(\beta_{6}\mathstrut -\mathstrut \) \(156\) \(\beta_{5}\mathstrut -\mathstrut \) \(386\) \(\beta_{4}\mathstrut -\mathstrut \) \(338\) \(\beta_{3}\mathstrut +\mathstrut \) \(838\) \(\beta_{2}\mathstrut +\mathstrut \) \(154\) \(\beta_{1}\mathstrut +\mathstrut \) \(1580\)
\(\nu^{9}\)\(=\)\(-\)\(1138\) \(\beta_{19}\mathstrut -\mathstrut \) \(2219\) \(\beta_{18}\mathstrut -\mathstrut \) \(143\) \(\beta_{17}\mathstrut -\mathstrut \) \(1172\) \(\beta_{16}\mathstrut +\mathstrut \) \(2132\) \(\beta_{15}\mathstrut +\mathstrut \) \(66\) \(\beta_{14}\mathstrut -\mathstrut \) \(293\) \(\beta_{13}\mathstrut -\mathstrut \) \(2735\) \(\beta_{12}\mathstrut +\mathstrut \) \(629\) \(\beta_{11}\mathstrut -\mathstrut \) \(1507\) \(\beta_{10}\mathstrut +\mathstrut \) \(767\) \(\beta_{9}\mathstrut +\mathstrut \) \(2669\) \(\beta_{8}\mathstrut -\mathstrut \) \(210\) \(\beta_{7}\mathstrut -\mathstrut \) \(203\) \(\beta_{6}\mathstrut +\mathstrut \) \(241\) \(\beta_{5}\mathstrut -\mathstrut \) \(765\) \(\beta_{4}\mathstrut -\mathstrut \) \(1280\) \(\beta_{3}\mathstrut +\mathstrut \) \(2058\) \(\beta_{2}\mathstrut +\mathstrut \) \(829\) \(\beta_{1}\mathstrut +\mathstrut \) \(2822\)
\(\nu^{10}\)\(=\)\(-\)\(1264\) \(\beta_{19}\mathstrut -\mathstrut \) \(4473\) \(\beta_{18}\mathstrut +\mathstrut \) \(531\) \(\beta_{17}\mathstrut -\mathstrut \) \(5392\) \(\beta_{16}\mathstrut +\mathstrut \) \(2636\) \(\beta_{15}\mathstrut -\mathstrut \) \(1008\) \(\beta_{14}\mathstrut +\mathstrut \) \(1313\) \(\beta_{13}\mathstrut -\mathstrut \) \(7149\) \(\beta_{12}\mathstrut +\mathstrut \) \(2702\) \(\beta_{11}\mathstrut -\mathstrut \) \(3859\) \(\beta_{10}\mathstrut +\mathstrut \) \(1416\) \(\beta_{9}\mathstrut +\mathstrut \) \(5184\) \(\beta_{8}\mathstrut +\mathstrut \) \(450\) \(\beta_{7}\mathstrut -\mathstrut \) \(2363\) \(\beta_{6}\mathstrut -\mathstrut \) \(1648\) \(\beta_{5}\mathstrut -\mathstrut \) \(4529\) \(\beta_{4}\mathstrut -\mathstrut \) \(3968\) \(\beta_{3}\mathstrut +\mathstrut \) \(8781\) \(\beta_{2}\mathstrut +\mathstrut \) \(1619\) \(\beta_{1}\mathstrut +\mathstrut \) \(15866\)
\(\nu^{11}\)\(=\)\(-\)\(10969\) \(\beta_{19}\mathstrut -\mathstrut \) \(24806\) \(\beta_{18}\mathstrut -\mathstrut \) \(568\) \(\beta_{17}\mathstrut -\mathstrut \) \(15248\) \(\beta_{16}\mathstrut +\mathstrut \) \(21386\) \(\beta_{15}\mathstrut +\mathstrut \) \(232\) \(\beta_{14}\mathstrut -\mathstrut \) \(2794\) \(\beta_{13}\mathstrut -\mathstrut \) \(30504\) \(\beta_{12}\mathstrut +\mathstrut \) \(8314\) \(\beta_{11}\mathstrut -\mathstrut \) \(16875\) \(\beta_{10}\mathstrut +\mathstrut \) \(8703\) \(\beta_{9}\mathstrut +\mathstrut \) \(28497\) \(\beta_{8}\mathstrut -\mathstrut \) \(2592\) \(\beta_{7}\mathstrut -\mathstrut \) \(3210\) \(\beta_{6}\mathstrut +\mathstrut \) \(318\) \(\beta_{5}\mathstrut -\mathstrut \) \(10698\) \(\beta_{4}\mathstrut -\mathstrut \) \(14317\) \(\beta_{3}\mathstrut +\mathstrut \) \(24090\) \(\beta_{2}\mathstrut +\mathstrut \) \(7484\) \(\beta_{1}\mathstrut +\mathstrut \) \(34970\)
\(\nu^{12}\)\(=\)\(-\)\(16762\) \(\beta_{19}\mathstrut -\mathstrut \) \(59503\) \(\beta_{18}\mathstrut +\mathstrut \) \(6425\) \(\beta_{17}\mathstrut -\mathstrut \) \(62221\) \(\beta_{16}\mathstrut +\mathstrut \) \(36441\) \(\beta_{15}\mathstrut -\mathstrut \) \(9021\) \(\beta_{14}\mathstrut +\mathstrut \) \(9268\) \(\beta_{13}\mathstrut -\mathstrut \) \(88075\) \(\beta_{12}\mathstrut +\mathstrut \) \(32710\) \(\beta_{11}\mathstrut -\mathstrut \) \(47652\) \(\beta_{10}\mathstrut +\mathstrut \) \(19320\) \(\beta_{9}\mathstrut +\mathstrut \) \(66799\) \(\beta_{8}\mathstrut +\mathstrut \) \(603\) \(\beta_{7}\mathstrut -\mathstrut \) \(24465\) \(\beta_{6}\mathstrut -\mathstrut \) \(17330\) \(\beta_{5}\mathstrut -\mathstrut \) \(52117\) \(\beta_{4}\mathstrut -\mathstrut \) \(46152\) \(\beta_{3}\mathstrut +\mathstrut \) \(94967\) \(\beta_{2}\mathstrut +\mathstrut \) \(17125\) \(\beta_{1}\mathstrut +\mathstrut \) \(166222\)
\(\nu^{13}\)\(=\)\(-\)\(108410\) \(\beta_{19}\mathstrut -\mathstrut \) \(276964\) \(\beta_{18}\mathstrut +\mathstrut \) \(2777\) \(\beta_{17}\mathstrut -\mathstrut \) \(187462\) \(\beta_{16}\mathstrut +\mathstrut \) \(220379\) \(\beta_{15}\mathstrut -\mathstrut \) \(1990\) \(\beta_{14}\mathstrut -\mathstrut \) \(25937\) \(\beta_{13}\mathstrut -\mathstrut \) \(343613\) \(\beta_{12}\mathstrut +\mathstrut \) \(103527\) \(\beta_{11}\mathstrut -\mathstrut \) \(189898\) \(\beta_{10}\mathstrut +\mathstrut \) \(96997\) \(\beta_{9}\mathstrut +\mathstrut \) \(309852\) \(\beta_{8}\mathstrut -\mathstrut \) \(31012\) \(\beta_{7}\mathstrut -\mathstrut \) \(44127\) \(\beta_{6}\mathstrut -\mathstrut \) \(14036\) \(\beta_{5}\mathstrut -\mathstrut \) \(138298\) \(\beta_{4}\mathstrut -\mathstrut \) \(162100\) \(\beta_{3}\mathstrut +\mathstrut \) \(279684\) \(\beta_{2}\mathstrut +\mathstrut \) \(72380\) \(\beta_{1}\mathstrut +\mathstrut \) \(422041\)
\(\nu^{14}\)\(=\)\(-\)\(209183\) \(\beta_{19}\mathstrut -\mathstrut \) \(744000\) \(\beta_{18}\mathstrut +\mathstrut \) \(74499\) \(\beta_{17}\mathstrut -\mathstrut \) \(711737\) \(\beta_{16}\mathstrut +\mathstrut \) \(467037\) \(\beta_{15}\mathstrut -\mathstrut \) \(82961\) \(\beta_{14}\mathstrut +\mathstrut \) \(60285\) \(\beta_{13}\mathstrut -\mathstrut \) \(1053534\) \(\beta_{12}\mathstrut +\mathstrut \) \(387155\) \(\beta_{11}\mathstrut -\mathstrut \) \(570673\) \(\beta_{10}\mathstrut +\mathstrut \) \(245448\) \(\beta_{9}\mathstrut +\mathstrut \) \(822878\) \(\beta_{8}\mathstrut -\mathstrut \) \(30804\) \(\beta_{7}\mathstrut -\mathstrut \) \(258815\) \(\beta_{6}\mathstrut -\mathstrut \) \(184405\) \(\beta_{5}\mathstrut -\mathstrut \) \(595783\) \(\beta_{4}\mathstrut -\mathstrut \) \(533965\) \(\beta_{3}\mathstrut +\mathstrut \) \(1047742\) \(\beta_{2}\mathstrut +\mathstrut \) \(184679\) \(\beta_{1}\mathstrut +\mathstrut \) \(1792079\)
\(\nu^{15}\)\(=\)\(-\)\(1104331\) \(\beta_{19}\mathstrut -\mathstrut \) \(3104435\) \(\beta_{18}\mathstrut +\mathstrut \) \(105587\) \(\beta_{17}\mathstrut -\mathstrut \) \(2236833\) \(\beta_{16}\mathstrut +\mathstrut \) \(2330442\) \(\beta_{15}\mathstrut -\mathstrut \) \(61043\) \(\beta_{14}\mathstrut -\mathstrut \) \(241744\) \(\beta_{13}\mathstrut -\mathstrut \) \(3894167\) \(\beta_{12}\mathstrut +\mathstrut \) \(1247950\) \(\beta_{11}\mathstrut -\mathstrut \) \(2146634\) \(\beta_{10}\mathstrut +\mathstrut \) \(1080512\) \(\beta_{9}\mathstrut +\mathstrut \) \(3420549\) \(\beta_{8}\mathstrut -\mathstrut \) \(364106\) \(\beta_{7}\mathstrut -\mathstrut \) \(565284\) \(\beta_{6}\mathstrut -\mathstrut \) \(288629\) \(\beta_{5}\mathstrut -\mathstrut \) \(1708997\) \(\beta_{4}\mathstrut -\mathstrut \) \(1846347\) \(\beta_{3}\mathstrut +\mathstrut \) \(3231813\) \(\beta_{2}\mathstrut +\mathstrut \) \(735107\) \(\beta_{1}\mathstrut +\mathstrut \) \(5004206\)
\(\nu^{16}\)\(=\)\(-\)\(2526122\) \(\beta_{19}\mathstrut -\mathstrut \) \(8992100\) \(\beta_{18}\mathstrut +\mathstrut \) \(851497\) \(\beta_{17}\mathstrut -\mathstrut \) \(8125784\) \(\beta_{16}\mathstrut +\mathstrut \) \(5737033\) \(\beta_{15}\mathstrut -\mathstrut \) \(791883\) \(\beta_{14}\mathstrut +\mathstrut \) \(336048\) \(\beta_{13}\mathstrut -\mathstrut \) \(12388225\) \(\beta_{12}\mathstrut +\mathstrut \) \(4520141\) \(\beta_{11}\mathstrut -\mathstrut \) \(6716596\) \(\beta_{10}\mathstrut +\mathstrut \) \(2996743\) \(\beta_{9}\mathstrut +\mathstrut \) \(9866095\) \(\beta_{8}\mathstrut -\mathstrut \) \(651710\) \(\beta_{7}\mathstrut -\mathstrut \) \(2789887\) \(\beta_{6}\mathstrut -\mathstrut \) \(1993790\) \(\beta_{5}\mathstrut -\mathstrut \) \(6796932\) \(\beta_{4}\mathstrut -\mathstrut \) \(6155459\) \(\beta_{3}\mathstrut +\mathstrut \) \(11707236\) \(\beta_{2}\mathstrut +\mathstrut \) \(2030098\) \(\beta_{1}\mathstrut +\mathstrut \) \(19702435\)
\(\nu^{17}\)\(=\)\(-\)\(11579025\) \(\beta_{19}\mathstrut -\mathstrut \) \(34964785\) \(\beta_{18}\mathstrut +\mathstrut \) \(1772647\) \(\beta_{17}\mathstrut -\mathstrut \) \(26247096\) \(\beta_{16}\mathstrut +\mathstrut \) \(25185774\) \(\beta_{15}\mathstrut -\mathstrut \) \(1002953\) \(\beta_{14}\mathstrut -\mathstrut \) \(2299329\) \(\beta_{13}\mathstrut -\mathstrut \) \(44297032\) \(\beta_{12}\mathstrut +\mathstrut \) \(14759702\) \(\beta_{11}\mathstrut -\mathstrut \) \(24353161\) \(\beta_{10}\mathstrut +\mathstrut \) \(12093232\) \(\beta_{9}\mathstrut +\mathstrut \) \(38191290\) \(\beta_{8}\mathstrut -\mathstrut \) \(4225122\) \(\beta_{7}\mathstrut -\mathstrut \) \(6956509\) \(\beta_{6}\mathstrut -\mathstrut \) \(4233775\) \(\beta_{5}\mathstrut -\mathstrut \) \(20548142\) \(\beta_{4}\mathstrut -\mathstrut \) \(21089343\) \(\beta_{3}\mathstrut +\mathstrut \) \(37233500\) \(\beta_{2}\mathstrut +\mathstrut \) \(7737117\) \(\beta_{1}\mathstrut +\mathstrut \) \(58642246\)
\(\nu^{18}\)\(=\)\(-\)\(29926581\) \(\beta_{19}\mathstrut -\mathstrut \) \(106533226\) \(\beta_{18}\mathstrut +\mathstrut \) \(9696174\) \(\beta_{17}\mathstrut -\mathstrut \) \(92787727\) \(\beta_{16}\mathstrut +\mathstrut \) \(68710207\) \(\beta_{15}\mathstrut -\mathstrut \) \(7859457\) \(\beta_{14}\mathstrut +\mathstrut \) \(1145231\) \(\beta_{13}\mathstrut -\mathstrut \) \(144184079\) \(\beta_{12}\mathstrut +\mathstrut \) \(52341345\) \(\beta_{11}\mathstrut -\mathstrut \) \(78240298\) \(\beta_{10}\mathstrut +\mathstrut \) \(35736779\) \(\beta_{9}\mathstrut +\mathstrut \) \(116327848\) \(\beta_{8}\mathstrut -\mathstrut \) \(9777275\) \(\beta_{7}\mathstrut -\mathstrut \) \(30538319\) \(\beta_{6}\mathstrut -\mathstrut \) \(21882293\) \(\beta_{5}\mathstrut -\mathstrut \) \(77517896\) \(\beta_{4}\mathstrut -\mathstrut \) \(70781812\) \(\beta_{3}\mathstrut +\mathstrut \) \(131895213\) \(\beta_{2}\mathstrut +\mathstrut \) \(22648826\) \(\beta_{1}\mathstrut +\mathstrut \) \(219512538\)
\(\nu^{19}\)\(=\)\(-\)\(124401332\) \(\beta_{19}\mathstrut -\mathstrut \) \(395489373\) \(\beta_{18}\mathstrut +\mathstrut \) \(24516859\) \(\beta_{17}\mathstrut -\mathstrut \) \(304997153\) \(\beta_{16}\mathstrut +\mathstrut \) \(276834042\) \(\beta_{15}\mathstrut -\mathstrut \) \(13854486\) \(\beta_{14}\mathstrut -\mathstrut \) \(22495592\) \(\beta_{13}\mathstrut -\mathstrut \) \(505056844\) \(\beta_{12}\mathstrut +\mathstrut \) \(172541993\) \(\beta_{11}\mathstrut -\mathstrut \) \(277024285\) \(\beta_{10}\mathstrut +\mathstrut \) \(136085690\) \(\beta_{9}\mathstrut +\mathstrut \) \(429863620\) \(\beta_{8}\mathstrut -\mathstrut \) \(48683772\) \(\beta_{7}\mathstrut -\mathstrut \) \(83531839\) \(\beta_{6}\mathstrut -\mathstrut \) \(55250151\) \(\beta_{5}\mathstrut -\mathstrut \) \(242850258\) \(\beta_{4}\mathstrut -\mathstrut \) \(241196625\) \(\beta_{3}\mathstrut +\mathstrut \) \(428118285\) \(\beta_{2}\mathstrut +\mathstrut \) \(83567171\) \(\beta_{1}\mathstrut +\mathstrut \) \(681858800\)

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
3.38332
2.69870
2.25007
2.11393
1.80437
1.37441
1.06424
0.609415
0.283688
0.237078
−0.126023
−0.313711
−0.423017
−1.12918
−1.36564
−1.42163
−1.50131
−2.06684
−2.70148
−2.77040
0 −3.38332 0 −0.369303 0 0.374879 0 8.44685 0
1.2 0 −2.69870 0 −1.98742 0 −0.0153109 0 4.28301 0
1.3 0 −2.25007 0 1.77999 0 3.34275 0 2.06282 0
1.4 0 −2.11393 0 0.387846 0 2.46344 0 1.46870 0
1.5 0 −1.80437 0 3.22329 0 2.17270 0 0.255750 0
1.6 0 −1.37441 0 −2.22841 0 2.74311 0 −1.11098 0
1.7 0 −1.06424 0 −1.91818 0 −4.17854 0 −1.86738 0
1.8 0 −0.609415 0 0.676769 0 −1.71345 0 −2.62861 0
1.9 0 −0.283688 0 −2.93013 0 4.78235 0 −2.91952 0
1.10 0 −0.237078 0 2.74912 0 −2.88085 0 −2.94379 0
1.11 0 0.126023 0 −3.00184 0 1.14075 0 −2.98412 0
1.12 0 0.313711 0 3.13839 0 2.81435 0 −2.90159 0
1.13 0 0.423017 0 1.16659 0 0.651110 0 −2.82106 0
1.14 0 1.12918 0 −2.32377 0 1.49770 0 −1.72496 0
1.15 0 1.36564 0 2.75901 0 −0.969675 0 −1.13502 0
1.16 0 1.42163 0 −3.52594 0 −1.45144 0 −0.978971 0
1.17 0 1.50131 0 1.01639 0 −3.17746 0 −0.746069 0
1.18 0 2.06684 0 0.0661094 0 2.71031 0 1.27184 0
1.19 0 2.70148 0 −2.07497 0 −1.35978 0 4.29801 0
1.20 0 2.77040 0 1.39646 0 −3.94694 0 4.67511 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(-1\)
\(59\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8024))\):

\(T_{3}^{20} + \cdots\)
\(T_{5}^{20} + \cdots\)
\(T_{7}^{20} - \cdots\)